Theorem ifnebib 32750

Description: The converse of ifbi 4505 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 4524	. . 3 ⊢ (if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵) ↔ ((𝜓 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴) ∨ (¬ 𝜓 ∧ if(𝜑, 𝐴, 𝐵) = 𝐵)))
2		ifnetrue 32748	. . . . . 6 ⊢ ((𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴) → 𝜑)
3	2	adantrl 726	. . . . 5 ⊢ ((𝐴 ≠ 𝐵 ∧ (𝜓 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴)) → 𝜑)
4		simprl 780	. . . . 5 ⊢ ((𝐴 ≠ 𝐵 ∧ (𝜓 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴)) → 𝜓)
5	3, 4	2thd 267	. . . 4 ⊢ ((𝐴 ≠ 𝐵 ∧ (𝜓 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴)) → (𝜑 ↔ 𝜓))
6		ifnefals 32749	. . . . . 6 ⊢ ((𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐵) → ¬ 𝜑)
7	6	adantrl 726	. . . . 5 ⊢ ((𝐴 ≠ 𝐵 ∧ (¬ 𝜓 ∧ if(𝜑, 𝐴, 𝐵) = 𝐵)) → ¬ 𝜑)
8		simprl 780	. . . . 5 ⊢ ((𝐴 ≠ 𝐵 ∧ (¬ 𝜓 ∧ if(𝜑, 𝐴, 𝐵) = 𝐵)) → ¬ 𝜓)
9	7, 8	2falsed 378	. . . 4 ⊢ ((𝐴 ≠ 𝐵 ∧ (¬ 𝜓 ∧ if(𝜑, 𝐴, 𝐵) = 𝐵)) → (𝜑 ↔ 𝜓))
10	5, 9	jaodan 970	. . 3 ⊢ ((𝐴 ≠ 𝐵 ∧ ((𝜓 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴) ∨ (¬ 𝜓 ∧ if(𝜑, 𝐴, 𝐵) = 𝐵))) → (𝜑 ↔ 𝜓))
11	1, 10	sylan2b 603	. 2 ⊢ ((𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵)) → (𝜑 ↔ 𝜓))
12		ifbi 4505	. . 3 ⊢ ((𝜑 ↔ 𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))
13	12	adantl 485	. 2 ⊢ ((𝐴 ≠ 𝐵 ∧ (𝜑 ↔ 𝜓)) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))
14	11, 13	impbida 810	1 ⊢ (𝐴 ≠ 𝐵 → (if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵) ↔ (𝜑 ↔ 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   ∨ wo 858   = wceq 1562   ≠ wne 2959  ifcif 4482

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ne 2960  df-if 4483

This theorem is referenced by:  ply1moneq  33786