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Theorem necon3bd 2298
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 596 . 2 (𝜑 → (¬ 𝜓 → ¬ 𝐴 = 𝐵))
3 df-ne 2256 . 2 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
42, 3syl6ibr 160 1 (𝜑 → (¬ 𝜓𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1289  wne 2255
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580
This theorem depends on definitions:  df-bi 115  df-ne 2256
This theorem is referenced by:  nelne1  2345  nelne2  2346  nssne1  3080  nssne2  3081  disjne  3333  difsn  3569  nbrne1  3854  nbrne2  3855  ac6sfi  6594  indpi  6880  zneo  8817
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