Theorem ifnefals

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

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

Proof

Step	Hyp	Ref	Expression
1		iftrue	⊢ ( 𝜑 → if ( 𝜑 , 𝐴 , 𝐵 ) = 𝐴 )
2	1	adantl	⊢ ( ( ( 𝐴 ≠ 𝐵 ∧ if ( 𝜑 , 𝐴 , 𝐵 ) = 𝐵 ) ∧ 𝜑 ) → if ( 𝜑 , 𝐴 , 𝐵 ) = 𝐴 )
3		simplr	⊢ ( ( ( 𝐴 ≠ 𝐵 ∧ if ( 𝜑 , 𝐴 , 𝐵 ) = 𝐵 ) ∧ 𝜑 ) → if ( 𝜑 , 𝐴 , 𝐵 ) = 𝐵 )
4		simpll	⊢ ( ( ( 𝐴 ≠ 𝐵 ∧ if ( 𝜑 , 𝐴 , 𝐵 ) = 𝐵 ) ∧ 𝜑 ) → 𝐴 ≠ 𝐵 )
5	4	necomd	⊢ ( ( ( 𝐴 ≠ 𝐵 ∧ if ( 𝜑 , 𝐴 , 𝐵 ) = 𝐵 ) ∧ 𝜑 ) → 𝐵 ≠ 𝐴 )
6	3 5	eqnetrd	⊢ ( ( ( 𝐴 ≠ 𝐵 ∧ if ( 𝜑 , 𝐴 , 𝐵 ) = 𝐵 ) ∧ 𝜑 ) → if ( 𝜑 , 𝐴 , 𝐵 ) ≠ 𝐴 )
7	6	neneqd	⊢ ( ( ( 𝐴 ≠ 𝐵 ∧ if ( 𝜑 , 𝐴 , 𝐵 ) = 𝐵 ) ∧ 𝜑 ) → ¬ if ( 𝜑 , 𝐴 , 𝐵 ) = 𝐴 )
8	2 7	pm2.65da	⊢ ( ( 𝐴 ≠ 𝐵 ∧ if ( 𝜑 , 𝐴 , 𝐵 ) = 𝐵 ) → ¬ 𝜑 )