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Metamath Proof Explorer

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 )