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Theorem ifnebib 32570
Description: The converse of ifbi 4553 holds if the two values are not equal. (Contributed by Thierry Arnoux, 20-Feb-2025.)
Assertion
Ref Expression
ifnebib (𝐴𝐵 → (if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵) ↔ (𝜑𝜓)))

Proof of Theorem ifnebib
StepHypRef Expression
1 eqif 4572 . . 3 (if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵) ↔ ((𝜓 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴) ∨ (¬ 𝜓 ∧ if(𝜑, 𝐴, 𝐵) = 𝐵)))
2 ifnetrue 32568 . . . . . 6 ((𝐴𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴) → 𝜑)
32adantrl 716 . . . . 5 ((𝐴𝐵 ∧ (𝜓 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴)) → 𝜑)
4 simprl 771 . . . . 5 ((𝐴𝐵 ∧ (𝜓 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴)) → 𝜓)
53, 42thd 265 . . . 4 ((𝐴𝐵 ∧ (𝜓 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴)) → (𝜑𝜓))
6 ifnefals 32569 . . . . . 6 ((𝐴𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐵) → ¬ 𝜑)
76adantrl 716 . . . . 5 ((𝐴𝐵 ∧ (¬ 𝜓 ∧ if(𝜑, 𝐴, 𝐵) = 𝐵)) → ¬ 𝜑)
8 simprl 771 . . . . 5 ((𝐴𝐵 ∧ (¬ 𝜓 ∧ if(𝜑, 𝐴, 𝐵) = 𝐵)) → ¬ 𝜓)
97, 82falsed 376 . . . 4 ((𝐴𝐵 ∧ (¬ 𝜓 ∧ if(𝜑, 𝐴, 𝐵) = 𝐵)) → (𝜑𝜓))
105, 9jaodan 959 . . 3 ((𝐴𝐵 ∧ ((𝜓 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴) ∨ (¬ 𝜓 ∧ if(𝜑, 𝐴, 𝐵) = 𝐵))) → (𝜑𝜓))
111, 10sylan2b 594 . 2 ((𝐴𝐵 ∧ if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵)) → (𝜑𝜓))
12 ifbi 4553 . . 3 ((𝜑𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))
1312adantl 481 . 2 ((𝐴𝐵 ∧ (𝜑𝜓)) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))
1411, 13impbida 801 1 (𝐴𝐵 → (if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵) ↔ (𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847   = wceq 1537  wne 2938  ifcif 4531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-if 4532
This theorem is referenced by:  ply1moneq  33591
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