Theorem ifnetrue

Description: Deduce truth from a conditional operator value. (Contributed by Thierry Arnoux, 20-Feb-2025)

	Ref	Expression
Assertion	ifnetrue	⊢ ( ( 𝐴 ≠ 𝐵 ∧ if ( 𝜑 , 𝐴 , 𝐵 ) = 𝐴 ) → 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		iffalse	⊢ ( ¬ 𝜑 → if ( 𝜑 , 𝐴 , 𝐵 ) = 𝐵 )
2	1	adantl	⊢ ( ( ( 𝐴 ≠ 𝐵 ∧ if ( 𝜑 , 𝐴 , 𝐵 ) = 𝐴 ) ∧ ¬ 𝜑 ) → if ( 𝜑 , 𝐴 , 𝐵 ) = 𝐵 )
3		simplr	⊢ ( ( ( 𝐴 ≠ 𝐵 ∧ if ( 𝜑 , 𝐴 , 𝐵 ) = 𝐴 ) ∧ ¬ 𝜑 ) → if ( 𝜑 , 𝐴 , 𝐵 ) = 𝐴 )
4		simpll	⊢ ( ( ( 𝐴 ≠ 𝐵 ∧ if ( 𝜑 , 𝐴 , 𝐵 ) = 𝐴 ) ∧ ¬ 𝜑 ) → 𝐴 ≠ 𝐵 )
5	3 4	eqnetrd	⊢ ( ( ( 𝐴 ≠ 𝐵 ∧ if ( 𝜑 , 𝐴 , 𝐵 ) = 𝐴 ) ∧ ¬ 𝜑 ) → if ( 𝜑 , 𝐴 , 𝐵 ) ≠ 𝐵 )
6	5	neneqd	⊢ ( ( ( 𝐴 ≠ 𝐵 ∧ if ( 𝜑 , 𝐴 , 𝐵 ) = 𝐴 ) ∧ ¬ 𝜑 ) → ¬ if ( 𝜑 , 𝐴 , 𝐵 ) = 𝐵 )
7	2 6	condan	⊢ ( ( 𝐴 ≠ 𝐵 ∧ if ( 𝜑 , 𝐴 , 𝐵 ) = 𝐴 ) → 𝜑 )