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Theorem neneqad 2364
Description: If it is not the case that two classes are equal, they are unequal. Converse of neneqd 2306. One-way deduction form of df-ne 2286. (Contributed by David Moews, 28-Feb-2017.)
Hypothesis
Ref Expression
neneqad.1 (𝜑 → ¬ 𝐴 = 𝐵)
Assertion
Ref Expression
neneqad (𝜑𝐴𝐵)

Proof of Theorem neneqad
StepHypRef Expression
1 neneqad.1 . . 3 (𝜑 → ¬ 𝐴 = 𝐵)
21con2i 601 . 2 (𝐴 = 𝐵 → ¬ 𝜑)
32necon2ai 2339 1 (𝜑𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1316  wne 2285
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 588  ax-in2 589
This theorem depends on definitions:  df-bi 116  df-ne 2286
This theorem is referenced by:  ne0i  3339  nsuceq0g  4310  fidifsnen  6732  nqnq0pi  7214  xrlttri3  9551
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