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Theorem necon3ad 2389
Description: Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.)
Hypothesis
Ref Expression
necon3ad.1 (𝜑 → (𝜓𝐴 = 𝐵))
Assertion
Ref Expression
necon3ad (𝜑 → (𝐴𝐵 → ¬ 𝜓))

Proof of Theorem necon3ad
StepHypRef Expression
1 df-ne 2348 . 2 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
2 necon3ad.1 . . 3 (𝜑 → (𝜓𝐴 = 𝐵))
32con3d 631 . 2 (𝜑 → (¬ 𝐴 = 𝐵 → ¬ 𝜓))
41, 3biimtrid 152 1 (𝜑 → (𝐴𝐵 → ¬ 𝜓))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1353  wne 2347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-in1 614  ax-in2 615
This theorem depends on definitions:  df-bi 117  df-ne 2348
This theorem is referenced by:  necon3d  2391  disjnim  3991  fodjumkvlemres  7150  nlt1pig  7318  eucalglt  12027  nprm  12093  pcprmpw2  12302  pcmpt  12311  expnprm  12321  0nnei  13286  2sqlem8a  14091  bj-charfunr  14184
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