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Theorem necon3bd 2242
 Description: Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.)
Hypothesis
Ref Expression
necon3bd.1 (φ → (A = Bψ))
Assertion
Ref Expression
necon3bd (φ → (¬ ψAB))

Proof of Theorem necon3bd
StepHypRef Expression
1 necon3bd.1 . . 3 (φ → (A = Bψ))
21con3d 560 . 2 (φ → (¬ ψ → ¬ A = B))
3 df-ne 2203 . 2 (AB ↔ ¬ A = B)
42, 3syl6ibr 151 1 (φ → (¬ ψAB))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1242   ≠ wne 2201 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 2203 This theorem is referenced by:  nelne1  2289  nelne2  2290  nssne1  2995  nssne2  2996  disjne  3267  difsn  3492  nbrne1  3772  nbrne2  3773  indpi  6326  zneo  8095
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