Theorem ifnefals 32568

Description: Deduce falsehood from a conditional operator value. (Contributed by Thierry Arnoux, 20-Feb-2025.)

Assertion

Ref	Expression
ifnefals	⊢ ((𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐵) → ¬ 𝜑)

Proof of Theorem ifnefals

Step	Hyp	Ref	Expression
1		iftrue 4536	. . 3 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
2	1	adantl 481	. 2 ⊢ (((𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐵) ∧ 𝜑) → if(𝜑, 𝐴, 𝐵) = 𝐴)
3		simplr 769	. . . 4 ⊢ (((𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐵) ∧ 𝜑) → if(𝜑, 𝐴, 𝐵) = 𝐵)
4		simpll 767	. . . . 5 ⊢ (((𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐵) ∧ 𝜑) → 𝐴 ≠ 𝐵)
5	4	necomd 2993	. . . 4 ⊢ (((𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐵) ∧ 𝜑) → 𝐵 ≠ 𝐴)
6	3, 5	eqnetrd 3005	. . 3 ⊢ (((𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐵) ∧ 𝜑) → if(𝜑, 𝐴, 𝐵) ≠ 𝐴)
7	6	neneqd 2942	. 2 ⊢ (((𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐵) ∧ 𝜑) → ¬ if(𝜑, 𝐴, 𝐵) = 𝐴)
8	2, 7	pm2.65da 817	1 ⊢ ((𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐵) → ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1536   ≠ wne 2937  ifcif 4530

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-if 4531

This theorem is referenced by:  ifnebib  32569