Theorem neneqad 2459

Description: If it is not the case that two classes are equal, they are unequal. Converse of neneqd 2401. One-way deduction form of df-ne 2381. (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 630	. 2 ⊢ (𝐴 = 𝐵 → ¬ 𝜑)
3	2	necon2ai 2434	1 ⊢ (𝜑 → 𝐴 ≠ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1375   ≠ wne 2380

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618

This theorem depends on definitions:  df-bi 117  df-ne 2381

This theorem is referenced by:  ne0i  3478  nsuceq0g  4486  fidifsnen  7000  nqnq0pi  7593  xrlttri3  9961  lcmval  12551  lcmcllem  12555