Theorem ifnefals 33213

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 4491	. . 3 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
2	1	adantl 487	. 2 ⊢ (((𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐵) ∧ 𝜑) → if(𝜑, 𝐴, 𝐵) = 𝐴)
3		simplr 781	. . . 4 ⊢ (((𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐵) ∧ 𝜑) → if(𝜑, 𝐴, 𝐵) = 𝐵)
4		simpll 779	. . . . 5 ⊢ (((𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐵) ∧ 𝜑) → 𝐴 ≠ 𝐵)
5	4	necomd 3014	. . . 4 ⊢ (((𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐵) ∧ 𝜑) → 𝐵 ≠ 𝐴)
6	3, 5	eqnetrd 3026	. . 3 ⊢ (((𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐵) ∧ 𝜑) → if(𝜑, 𝐴, 𝐵) ≠ 𝐴)
7	6	neneqd 2964	. 2 ⊢ (((𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐵) ∧ 𝜑) → ¬ if(𝜑, 𝐴, 𝐵) = 𝐴)
8	2, 7	pm2.65da 829	1 ⊢ ((𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐵) → ¬ 𝜑)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ≠ wne 2959  ifcif 4485

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-if 4486

This theorem is used by:  ifnebib  33214