Theorem ifnefals 32636

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 4460	. . 3 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
2	1	adantl 482	. 2 ⊢ (((𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐵) ∧ 𝜑) → if(𝜑, 𝐴, 𝐵) = 𝐴)
3		simplr 774	. . . 4 ⊢ (((𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐵) ∧ 𝜑) → if(𝜑, 𝐴, 𝐵) = 𝐵)
4		simpll 772	. . . . 5 ⊢ (((𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐵) ∧ 𝜑) → 𝐴 ≠ 𝐵)
5	4	necomd 2989	. . . 4 ⊢ (((𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐵) ∧ 𝜑) → 𝐵 ≠ 𝐴)
6	3, 5	eqnetrd 3001	. . 3 ⊢ (((𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐵) ∧ 𝜑) → if(𝜑, 𝐴, 𝐵) ≠ 𝐴)
7	6	neneqd 2939	. 2 ⊢ (((𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐵) ∧ 𝜑) → ¬ if(𝜑, 𝐴, 𝐵) = 𝐴)
8	2, 7	pm2.65da 822	1 ⊢ ((𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐵) → ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1547   ≠ wne 2934  ifcif 4454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-if 4455

This theorem is referenced by:  ifnebib  32637