Theorem iftrueb 4493

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 2986	. . . . 5 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴)
2	1	biimpi 216	. . . 4 ⊢ (𝐴 ≠ 𝐵 → 𝐵 ≠ 𝐴)
3		iffalse 4489	. . . . 5 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
4	3	neeq1d 2992	. . . 4 ⊢ (¬ 𝜑 → (if(𝜑, 𝐴, 𝐵) ≠ 𝐴 ↔ 𝐵 ≠ 𝐴))
5	2, 4	syl5ibrcom 247	. . 3 ⊢ (𝐴 ≠ 𝐵 → (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) ≠ 𝐴))
6	5	necon4bd 2953	. 2 ⊢ (𝐴 ≠ 𝐵 → (if(𝜑, 𝐴, 𝐵) = 𝐴 → 𝜑))
7		iftrue 4486	. 2 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
8	6, 7	impbid1 225	1 ⊢ (𝐴 ≠ 𝐵 → (if(𝜑, 𝐴, 𝐵) = 𝐴 ↔ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   = wceq 1542   ≠ wne 2933  ifcif 4480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-if 4481

This theorem is referenced by:  psdmvr  22116