Theorem iftrueb 4499

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 3010	. . . . 5 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴)
2	1	biimpi 219	. . . 4 ⊢ (𝐴 ≠ 𝐵 → 𝐵 ≠ 𝐴)
3		iffalse 4495	. . . . 5 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
4	3	neeq1d 3016	. . . 4 ⊢ (¬ 𝜑 → (if(𝜑, 𝐴, 𝐵) ≠ 𝐴 ↔ 𝐵 ≠ 𝐴))
5	2, 4	syl5ibrcom 250	. . 3 ⊢ (𝐴 ≠ 𝐵 → (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) ≠ 𝐴))
6	5	necon4bd 2977	. 2 ⊢ (𝐴 ≠ 𝐵 → (if(𝜑, 𝐴, 𝐵) = 𝐴 → 𝜑))
7		iftrue 4492	. 2 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
8	6, 7	impbid1 228	1 ⊢ (𝐴 ≠ 𝐵 → (if(𝜑, 𝐴, 𝐵) = 𝐴 ↔ 𝜑))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   = wceq 1569   ≠ wne 2957  ifcif 4486

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-if 4487

This theorem is used by:  psdmvr  22343