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Theorem ifnetrue 32560
Description: Deduce truth from a conditional operator value. (Contributed by Thierry Arnoux, 20-Feb-2025.)
Assertion
Ref Expression
ifnetrue ((𝐴𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴) → 𝜑)

Proof of Theorem ifnetrue
StepHypRef Expression
1 iffalse 4534 . . 3 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
21adantl 481 . 2 (((𝐴𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴) ∧ ¬ 𝜑) → if(𝜑, 𝐴, 𝐵) = 𝐵)
3 simplr 769 . . . 4 (((𝐴𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴) ∧ ¬ 𝜑) → if(𝜑, 𝐴, 𝐵) = 𝐴)
4 simpll 767 . . . 4 (((𝐴𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴) ∧ ¬ 𝜑) → 𝐴𝐵)
53, 4eqnetrd 3008 . . 3 (((𝐴𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴) ∧ ¬ 𝜑) → if(𝜑, 𝐴, 𝐵) ≠ 𝐵)
65neneqd 2945 . 2 (((𝐴𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴) ∧ ¬ 𝜑) → ¬ if(𝜑, 𝐴, 𝐵) = 𝐵)
72, 6condan 818 1 ((𝐴𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴) → 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1540  wne 2940  ifcif 4525
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-if 4526
This theorem is referenced by:  ifnebib  32562
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