Theorem ifeq12da

Description: Equivalence deduction for conditional operators. (Contributed by Wolf Lammen, 24-Jun-2021)

	Ref	Expression
Hypotheses	ifeq12da.1	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝐴 = 𝐶 )
	ifeq12da.2	⊢ ( ( 𝜑 ∧ ¬ 𝜓 ) → 𝐵 = 𝐷 )
Assertion	ifeq12da	⊢ ( 𝜑 → if ( 𝜓 , 𝐴 , 𝐵 ) = if ( 𝜓 , 𝐶 , 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		ifeq12da.1	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝐴 = 𝐶 )
2		ifeq12da.2	⊢ ( ( 𝜑 ∧ ¬ 𝜓 ) → 𝐵 = 𝐷 )
3	1	ifeq1da	⊢ ( 𝜑 → if ( 𝜓 , 𝐴 , 𝐵 ) = if ( 𝜓 , 𝐶 , 𝐵 ) )
4		iftrue	⊢ ( 𝜓 → if ( 𝜓 , 𝐶 , 𝐵 ) = 𝐶 )
5		iftrue	⊢ ( 𝜓 → if ( 𝜓 , 𝐶 , 𝐷 ) = 𝐶 )
6	4 5	eqtr4d	⊢ ( 𝜓 → if ( 𝜓 , 𝐶 , 𝐵 ) = if ( 𝜓 , 𝐶 , 𝐷 ) )
7	3 6	sylan9eq	⊢ ( ( 𝜑 ∧ 𝜓 ) → if ( 𝜓 , 𝐴 , 𝐵 ) = if ( 𝜓 , 𝐶 , 𝐷 ) )
8	2	ifeq2da	⊢ ( 𝜑 → if ( 𝜓 , 𝐴 , 𝐵 ) = if ( 𝜓 , 𝐴 , 𝐷 ) )
9		iffalse	⊢ ( ¬ 𝜓 → if ( 𝜓 , 𝐴 , 𝐷 ) = 𝐷 )
10		iffalse	⊢ ( ¬ 𝜓 → if ( 𝜓 , 𝐶 , 𝐷 ) = 𝐷 )
11	9 10	eqtr4d	⊢ ( ¬ 𝜓 → if ( 𝜓 , 𝐴 , 𝐷 ) = if ( 𝜓 , 𝐶 , 𝐷 ) )
12	8 11	sylan9eq	⊢ ( ( 𝜑 ∧ ¬ 𝜓 ) → if ( 𝜓 , 𝐴 , 𝐵 ) = if ( 𝜓 , 𝐶 , 𝐷 ) )
13	7 12	pm2.61dan	⊢ ( 𝜑 → if ( 𝜓 , 𝐴 , 𝐵 ) = if ( 𝜓 , 𝐶 , 𝐷 ) )