Theorem ifnebib 32838

Description: The converse of ifbi 4515 holds if the two values are not equal. (Contributed by Thierry Arnoux, 20-Feb-2025.)

Assertion

Ref	Expression
ifnebib	⊢ (𝐴 ≠ 𝐵 → (if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵) ↔ (𝜑 ↔ 𝜓)))

Proof of Theorem ifnebib

Step	Hyp	Ref	Expression
1		eqif 4534	. . 3 ⊢ (if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵) ↔ ((𝜓 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴) ∨ (¬ 𝜓 ∧ if(𝜑, 𝐴, 𝐵) = 𝐵)))
2		ifnetrue 32836	. . . . . 6 ⊢ ((𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴) → 𝜑)
3	2	adantrl 728	. . . . 5 ⊢ ((𝐴 ≠ 𝐵 ∧ (𝜓 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴)) → 𝜑)
4		simprl 782	. . . . 5 ⊢ ((𝐴 ≠ 𝐵 ∧ (𝜓 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴)) → 𝜓)
5	3, 4	2thd 268	. . . 4 ⊢ ((𝐴 ≠ 𝐵 ∧ (𝜓 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴)) → (𝜑 ↔ 𝜓))
6		ifnefals 32837	. . . . . 6 ⊢ ((𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐵) → ¬ 𝜑)
7	6	adantrl 728	. . . . 5 ⊢ ((𝐴 ≠ 𝐵 ∧ (¬ 𝜓 ∧ if(𝜑, 𝐴, 𝐵) = 𝐵)) → ¬ 𝜑)
8		simprl 782	. . . . 5 ⊢ ((𝐴 ≠ 𝐵 ∧ (¬ 𝜓 ∧ if(𝜑, 𝐴, 𝐵) = 𝐵)) → ¬ 𝜓)
9	7, 8	2falsed 379	. . . 4 ⊢ ((𝐴 ≠ 𝐵 ∧ (¬ 𝜓 ∧ if(𝜑, 𝐴, 𝐵) = 𝐵)) → (𝜑 ↔ 𝜓))
10	5, 9	jaodan 972	. . 3 ⊢ ((𝐴 ≠ 𝐵 ∧ ((𝜓 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴) ∨ (¬ 𝜓 ∧ if(𝜑, 𝐴, 𝐵) = 𝐵))) → (𝜑 ↔ 𝜓))
11	1, 10	sylan2b 605	. 2 ⊢ ((𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵)) → (𝜑 ↔ 𝜓))
12		ifbi 4515	. . 3 ⊢ ((𝜑 ↔ 𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))
13	12	adantl 486	. 2 ⊢ ((𝐴 ≠ 𝐵 ∧ (𝜑 ↔ 𝜓)) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))
14	11, 13	impbida 812	1 ⊢ (𝐴 ≠ 𝐵 → (if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵) ↔ (𝜑 ↔ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1567   ≠ wne 2964  ifcif 4492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-if 4493

This theorem is referenced by:  ply1moneq  33825