Theorem ifid

Description: Identical true and false arguments in the conditional operator. (Contributed by NM, 18-Apr-2005)

	Ref	Expression
Assertion	ifid	⊢ if ( 𝜑 , 𝐴 , 𝐴 ) = 𝐴

Proof

Step	Hyp	Ref	Expression
1		iftrue	⊢ ( 𝜑 → if ( 𝜑 , 𝐴 , 𝐴 ) = 𝐴 )
2		iffalse	⊢ ( ¬ 𝜑 → if ( 𝜑 , 𝐴 , 𝐴 ) = 𝐴 )
3	1 2	pm2.61i	⊢ if ( 𝜑 , 𝐴 , 𝐴 ) = 𝐴