Theorem iftrueb 4494

Description: When the branches are not equal, an "if" condition results in the first branch if and only if its condition is true. (Contributed by SN, 16-Oct-2025.)

Assertion

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

Proof of Theorem iftrueb

Step	Hyp	Ref	Expression
1		necom 3011	. . . . 5 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴)
2	1	biimpi 218	. . . 4 ⊢ (𝐴 ≠ 𝐵 → 𝐵 ≠ 𝐴)
3		iffalse 4490	. . . . 5 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
4	3	neeq1d 3017	. . . 4 ⊢ (¬ 𝜑 → (if(𝜑, 𝐴, 𝐵) ≠ 𝐴 ↔ 𝐵 ≠ 𝐴))
5	2, 4	syl5ibrcom 249	. . 3 ⊢ (𝐴 ≠ 𝐵 → (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) ≠ 𝐴))
6	5	necon4bd 2978	. 2 ⊢ (𝐴 ≠ 𝐵 → (if(𝜑, 𝐴, 𝐵) = 𝐴 → 𝜑))
7		iftrue 4487	. 2 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
8	6, 7	impbid1 227	1 ⊢ (𝐴 ≠ 𝐵 → (if(𝜑, 𝐴, 𝐵) = 𝐴 ↔ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   = wceq 1561   ≠ wne 2958  ifcif 4481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-ex 1801  df-sb 2092  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-if 4482

This theorem is referenced by:  psdmvr  22241