Theorem ifeq2da

Description: Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009)

	Ref	Expression
Hypothesis	ifeq2da.1	⊢ ( ( 𝜑 ∧ ¬ 𝜓 ) → 𝐴 = 𝐵 )
Assertion	ifeq2da	⊢ ( 𝜑 → if ( 𝜓 , 𝐶 , 𝐴 ) = if ( 𝜓 , 𝐶 , 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		ifeq2da.1	⊢ ( ( 𝜑 ∧ ¬ 𝜓 ) → 𝐴 = 𝐵 )
2		iftrue	⊢ ( 𝜓 → if ( 𝜓 , 𝐶 , 𝐴 ) = 𝐶 )
3		iftrue	⊢ ( 𝜓 → if ( 𝜓 , 𝐶 , 𝐵 ) = 𝐶 )
4	2 3	eqtr4d	⊢ ( 𝜓 → if ( 𝜓 , 𝐶 , 𝐴 ) = if ( 𝜓 , 𝐶 , 𝐵 ) )
5	4	adantl	⊢ ( ( 𝜑 ∧ 𝜓 ) → if ( 𝜓 , 𝐶 , 𝐴 ) = if ( 𝜓 , 𝐶 , 𝐵 ) )
6	1	ifeq2d	⊢ ( ( 𝜑 ∧ ¬ 𝜓 ) → if ( 𝜓 , 𝐶 , 𝐴 ) = if ( 𝜓 , 𝐶 , 𝐵 ) )
7	5 6	pm2.61dan	⊢ ( 𝜑 → if ( 𝜓 , 𝐶 , 𝐴 ) = if ( 𝜓 , 𝐶 , 𝐵 ) )