Theorem ifnetrue 32803

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

Assertion

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

Proof of Theorem ifnetrue

Step	Hyp	Ref	Expression
1		iffalse 4492	. . 3 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
2	1	adantl 486	. 2 ⊢ (((𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴) ∧ ¬ 𝜑) → if(𝜑, 𝐴, 𝐵) = 𝐵)
3		simplr 780	. . . 4 ⊢ (((𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴) ∧ ¬ 𝜑) → if(𝜑, 𝐴, 𝐵) = 𝐴)
4		simpll 778	. . . 4 ⊢ (((𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴) ∧ ¬ 𝜑) → 𝐴 ≠ 𝐵)
5	3, 4	eqnetrd 3027	. . 3 ⊢ (((𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴) ∧ ¬ 𝜑) → if(𝜑, 𝐴, 𝐵) ≠ 𝐵)
6	5	neneqd 2965	. 2 ⊢ (((𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴) ∧ ¬ 𝜑) → ¬ if(𝜑, 𝐴, 𝐵) = 𝐵)
7	2, 6	condan 829	1 ⊢ ((𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴) → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1563   ≠ wne 2960  ifcif 4483

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-if 4484

This theorem is referenced by:  ifnebib  32805