Theorem neneqad 2419

Description: If it is not the case that two classes are equal, they are unequal. Converse of neneqd 2361. One-way deduction form of df-ne 2341. (Contributed by David Moews, 28-Feb-2017.)

Hypothesis

Ref	Expression
neneqad.1	⊢ (𝜑 → ¬ 𝐴 = 𝐵)

Assertion

Ref	Expression
neneqad	⊢ (𝜑 → 𝐴 ≠ 𝐵)

Proof of Theorem neneqad

Step	Hyp	Ref	Expression
1		neneqad.1	. . 3 ⊢ (𝜑 → ¬ 𝐴 = 𝐵)
2	1	con2i 622	. 2 ⊢ (𝐴 = 𝐵 → ¬ 𝜑)
3	2	necon2ai 2394	1 ⊢ (𝜑 → 𝐴 ≠ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1348   ≠ wne 2340

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610

This theorem depends on definitions:  df-bi 116  df-ne 2341

This theorem is referenced by:  ne0i  3421  nsuceq0g  4403  fidifsnen  6848  nqnq0pi  7400  xrlttri3  9754