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Theorem necon4bid 2823
Description: Contrapositive law deduction for inequality. (Contributed by NM, 29-Jun-2007.)
Hypothesis
Ref Expression
necon4bid.1 (𝜑 → (𝐴𝐵𝐶𝐷))
Assertion
Ref Expression
necon4bid (𝜑 → (𝐴 = 𝐵𝐶 = 𝐷))

Proof of Theorem necon4bid
StepHypRef Expression
1 necon4bid.1 . . 3 (𝜑 → (𝐴𝐵𝐶𝐷))
21necon2bbid 2821 . 2 (𝜑 → (𝐶 = 𝐷 ↔ ¬ 𝐴𝐵))
3 nne 2782 . 2 𝐴𝐵𝐴 = 𝐵)
42, 3syl6rbb 275 1 (𝜑 → (𝐴 = 𝐵𝐶 = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194   = wceq 1474  wne 2776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 195  df-ne 2778
This theorem is referenced by:  nebi  2858  znnenlem  14722  rpexp  15213  norm-i  27173  trlid0b  34283
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