Theorem ifeq1da

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

	Ref	Expression
Hypothesis	ifeq1da.1	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝐴 = 𝐵 )
Assertion	ifeq1da	⊢ ( 𝜑 → if ( 𝜓 , 𝐴 , 𝐶 ) = if ( 𝜓 , 𝐵 , 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		ifeq1da.1	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝐴 = 𝐵 )
2	1	ifeq1d	⊢ ( ( 𝜑 ∧ 𝜓 ) → if ( 𝜓 , 𝐴 , 𝐶 ) = if ( 𝜓 , 𝐵 , 𝐶 ) )
3		iffalse	⊢ ( ¬ 𝜓 → if ( 𝜓 , 𝐴 , 𝐶 ) = 𝐶 )
4		iffalse	⊢ ( ¬ 𝜓 → if ( 𝜓 , 𝐵 , 𝐶 ) = 𝐶 )
5	3 4	eqtr4d	⊢ ( ¬ 𝜓 → if ( 𝜓 , 𝐴 , 𝐶 ) = if ( 𝜓 , 𝐵 , 𝐶 ) )
6	5	adantl	⊢ ( ( 𝜑 ∧ ¬ 𝜓 ) → if ( 𝜓 , 𝐴 , 𝐶 ) = if ( 𝜓 , 𝐵 , 𝐶 ) )
7	2 6	pm2.61dan	⊢ ( 𝜑 → if ( 𝜓 , 𝐴 , 𝐶 ) = if ( 𝜓 , 𝐵 , 𝐶 ) )