Theorem ifnebib 33008

Description: The converse of ifbi 4508 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 4527	. . 3 ⊢ (if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵) ↔ ((𝜓 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴) ∨ (¬ 𝜓 ∧ if(𝜑, 𝐴, 𝐵) = 𝐵)))
2		ifnetrue 33006	. . . . . 6 ⊢ ((𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴) → 𝜑)
3	2	adantrl 729	. . . . 5 ⊢ ((𝐴 ≠ 𝐵 ∧ (𝜓 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴)) → 𝜑)
4		simprl 783	. . . . 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 33007	. . . . . 6 ⊢ ((𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐵) → ¬ 𝜑)
7	6	adantrl 729	. . . . 5 ⊢ ((𝐴 ≠ 𝐵 ∧ (¬ 𝜓 ∧ if(𝜑, 𝐴, 𝐵) = 𝐵)) → ¬ 𝜑)
8		simprl 783	. . . . 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 606	. 2 ⊢ ((𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵)) → (𝜑 ↔ 𝜓))
12		ifbi 4508	. . 3 ⊢ ((𝜑 ↔ 𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))
13	12	adantl 487	. 2 ⊢ ((𝐴 ≠ 𝐵 ∧ (𝜑 ↔ 𝜓)) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))
14	11, 13	impbida 813	1 ⊢ (𝐴 ≠ 𝐵 → (if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵) ↔ (𝜑 ↔ 𝜓)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ≠ wne 2957  ifcif 4485

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-if 4486

This theorem is used by:  ply1moneq  33983