Theorem jaeqifi

Description: Inference combining four equality antecedents into one equality of conditional operators antecedent. (Contributed by BTernaryTau, 14-Jul-2026)

	Ref	Expression
Hypotheses	jaeqifi.1	⊢ ( 𝐴 = 𝐶 → 𝜑 )
	jaeqifi.2	⊢ ( 𝐴 = 𝐷 → 𝜑 )
	jaeqifi.3	⊢ ( 𝐵 = 𝐶 → 𝜑 )
	jaeqifi.4	⊢ ( 𝐵 = 𝐷 → 𝜑 )
Assertion	jaeqifi	⊢ ( if ( 𝜓 , 𝐴 , 𝐵 ) = if ( 𝜒 , 𝐶 , 𝐷 ) → 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		jaeqifi.1	⊢ ( 𝐴 = 𝐶 → 𝜑 )
2		jaeqifi.2	⊢ ( 𝐴 = 𝐷 → 𝜑 )
3		jaeqifi.3	⊢ ( 𝐵 = 𝐶 → 𝜑 )
4		jaeqifi.4	⊢ ( 𝐵 = 𝐷 → 𝜑 )
5		iftrue	⊢ ( 𝜓 → if ( 𝜓 , 𝐴 , 𝐵 ) = 𝐴 )
6		iftrue	⊢ ( 𝜒 → if ( 𝜒 , 𝐶 , 𝐷 ) = 𝐶 )
7	5 6	eqeqan12d	⊢ ( ( 𝜓 ∧ 𝜒 ) → ( if ( 𝜓 , 𝐴 , 𝐵 ) = if ( 𝜒 , 𝐶 , 𝐷 ) ↔ 𝐴 = 𝐶 ) )
8	7 1	biimtrdi	⊢ ( ( 𝜓 ∧ 𝜒 ) → ( if ( 𝜓 , 𝐴 , 𝐵 ) = if ( 𝜒 , 𝐶 , 𝐷 ) → 𝜑 ) )
9		iffalse	⊢ ( ¬ 𝜒 → if ( 𝜒 , 𝐶 , 𝐷 ) = 𝐷 )
10	5 9	eqeqan12d	⊢ ( ( 𝜓 ∧ ¬ 𝜒 ) → ( if ( 𝜓 , 𝐴 , 𝐵 ) = if ( 𝜒 , 𝐶 , 𝐷 ) ↔ 𝐴 = 𝐷 ) )
11	10 2	biimtrdi	⊢ ( ( 𝜓 ∧ ¬ 𝜒 ) → ( if ( 𝜓 , 𝐴 , 𝐵 ) = if ( 𝜒 , 𝐶 , 𝐷 ) → 𝜑 ) )
12		iffalse	⊢ ( ¬ 𝜓 → if ( 𝜓 , 𝐴 , 𝐵 ) = 𝐵 )
13	12 6	eqeqan12d	⊢ ( ( ¬ 𝜓 ∧ 𝜒 ) → ( if ( 𝜓 , 𝐴 , 𝐵 ) = if ( 𝜒 , 𝐶 , 𝐷 ) ↔ 𝐵 = 𝐶 ) )
14	13 3	biimtrdi	⊢ ( ( ¬ 𝜓 ∧ 𝜒 ) → ( if ( 𝜓 , 𝐴 , 𝐵 ) = if ( 𝜒 , 𝐶 , 𝐷 ) → 𝜑 ) )
15	12 9	eqeqan12d	⊢ ( ( ¬ 𝜓 ∧ ¬ 𝜒 ) → ( if ( 𝜓 , 𝐴 , 𝐵 ) = if ( 𝜒 , 𝐶 , 𝐷 ) ↔ 𝐵 = 𝐷 ) )
16	15 4	biimtrdi	⊢ ( ( ¬ 𝜓 ∧ ¬ 𝜒 ) → ( if ( 𝜓 , 𝐴 , 𝐵 ) = if ( 𝜒 , 𝐶 , 𝐷 ) → 𝜑 ) )
17	8 11 14 16	4cases	⊢ ( if ( 𝜓 , 𝐴 , 𝐵 ) = if ( 𝜒 , 𝐶 , 𝐷 ) → 𝜑 )