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Theorem neneqad 2387
Description: If it is not the case that two classes are equal, they are unequal. Converse of neneqd 2329. One-way deduction form of df-ne 2309. (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 616 . 2 (𝐴 = 𝐵 → ¬ 𝜑)
32necon2ai 2362 1 (𝜑𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1331  wne 2308
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 2309
This theorem is referenced by:  ne0i  3369  nsuceq0g  4340  fidifsnen  6764  nqnq0pi  7253  xrlttri3  9590
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