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Theorem neneqad 2284
Description: If it is not the case that two classes are equal, they are unequal. Converse of neneqd 2226. One-way deduction form of df-ne 2206. (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 557 . 2 (𝐴 = 𝐵 → ¬ 𝜑)
32necon2ai 2259 1 (𝜑𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1243  wne 2204
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545
This theorem depends on definitions:  df-bi 110  df-ne 2206
This theorem is referenced by:  ne0i  3230  nsuceq0g  4155  fidifsnen  6331  nqnq0pi  6536  xrlttri3  8718  expival  9257
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