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Theorem ifnetrue 32567
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 4539 . . 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 3005 . . 3 (((𝐴𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴) ∧ ¬ 𝜑) → if(𝜑, 𝐴, 𝐵) ≠ 𝐵)
65neneqd 2942 . 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 1536  wne 2937  ifcif 4530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-if 4531
This theorem is referenced by:  ifnebib  32569
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