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namespace WpfButtonPerformClick
{
  using System;
  using System.Collections.Generic;
  using System.Linq;
  using System.Windows;
  using System.Windows.Controls;
  
  using WpfButtonPerformClick.Extensions;

  /// <summary>
  /// MainWindow.xaml の相互作用ロジック
  /// </summary>
  public partial class MainWindow : Window
  {
    public MainWindow()
    {
      InitializeComponent();
    }

    private void btn1_Click(object sender, RoutedEventArgs e)
    {
      this.FindByName<Button>("btn2").PerformClick();
    }

    private void btn2_Click(object sender, RoutedEventArgs e)
    {
      DataContext = "Button 2 Clicked.";
    }
  }
}

<Window x:Class="WpfButtonPerformClick.MainWindow"
        xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
        xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml"
        xmlns:sys="clr-namespace:System;assembly=mscorlib"
        Title="MainWindow" 
        Height="350" 
        Width="525">
  
  <Window.DataContext>
    <sys:String>hello world</sys:String>
  </Window.DataContext>
  <Grid>
    <Grid.RowDefinitions>
      <RowDefinition Height="Auto" />
      <RowDefinition />
    </Grid.RowDefinitions>
    
    <StackPanel Grid.Row="0" Orientation="Horizontal">
      <StackPanel.Resources>
        <Style TargetType="Button">
          <Setter Property="Margin" Value="10" />
          <Setter Property="Height" Value="40" />
          <Setter Property="Width" Value="100" />
        </Style>
      </StackPanel.Resources>
      <Button Name="btn1" Content="Button 1" Click="btn1_Click" />
      <Button Name="btn2" Content="Button 2" Click="btn2_Click" />
    </StackPanel>
    
    <DockPanel Grid.Row="1" LastChildFill="True">
      <TextBlock Text="{Binding Path=.}" FontSize="40" TextAlignment="Center" VerticalAlignment="Center" />
    </DockPanel>
  </Grid>
</Window>

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;
using System.Windows;

namespace WpfButtonPerformClick.Extensions
{
  public static class FrameworkElementExtensions
  {
    public static T FindByName<T>(this FrameworkElement self, string name) where T : FrameworkElement
    {
      return self.FindName(name) as T;
    }
  }
}

namespace WpfButtonPerformClick.Extensions
{
  using System;
  using System.Collections.Generic;
  using System.Linq;
  using System.Windows.Automation.Peers;
  using System.Windows.Automation.Provider;
  using System.Windows.Controls;

  public static class ButtonExtensions
  {
    public static void PerformClick(this Button self)
    {
      //
      // WinFormsでいうPerformClickを実行.
      //   System.Windows.Automation.Peers
      //   System.Windows.Automation.Provider
      // 名前空間が必要。
      //
      // IInvokeProviderインターフェースは、UIAutomationProvider.dll
      // の参照設定が追加で必要となる。
      //
      var peer     = new ButtonAutomationPeer(self);
      var provider = peer.GetPattern(PatternInterface.Invoke) as IInvokeProvider;

      provider.Invoke();      
    }
  }
}