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Theorem necon1bbid 2982
 Description: Contrapositive inference for inequality. (Contributed by NM, 31-Jan-2008.)
Hypothesis
Ref Expression
necon1bbid.1 (𝜑 → (𝐴𝐵𝜓))
Assertion
Ref Expression
necon1bbid (𝜑 → (¬ 𝜓𝐴 = 𝐵))

Proof of Theorem necon1bbid
StepHypRef Expression
1 df-ne 2944 . . 3 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
2 necon1bbid.1 . . 3 (𝜑 → (𝐴𝐵𝜓))
31, 2syl5bbr 274 . 2 (𝜑 → (¬ 𝐴 = 𝐵𝜓))
43con1bid 344 1 (𝜑 → (¬ 𝜓𝐴 = 𝐵))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   = wceq 1631   ≠ wne 2943 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 197  df-ne 2944 This theorem is referenced by:  necon4abid  2983  blssioo  22818  metdstri  22874  rrxmvallem  23406  dchrpt  25213  lgsquad3  25333  eupth2lem2  27399  lkrpssN  34972  dochshpsat  37264
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