Theorem neneqad 2385

Description: If it is not the case that two classes are equal, they are unequal. Converse of neneqd 2327. One-way deduction form of df-ne 2307. (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 616	. 2 ⊢ (𝐴 = 𝐵 → ¬ 𝜑)
3	2	necon2ai 2360	1 ⊢ (𝜑 → 𝐴 ≠ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1331   ≠ wne 2306

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 603  ax-in2 604

This theorem depends on definitions:  df-bi 116  df-ne 2307

This theorem is referenced by:  ne0i  3364  nsuceq0g  4335  fidifsnen  6757  nqnq0pi  7239  xrlttri3  9576