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Theorem necon1bbid 3026
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 2988 . . 3 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
2 necon1bbid.1 . . 3 (𝜑 → (𝐴𝐵𝜓))
31, 2bitr3id 288 . 2 (𝜑 → (¬ 𝐴 = 𝐵𝜓))
43con1bid 359 1 (𝜑 → (¬ 𝜓𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209   = wceq 1538  wne 2987
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 210  df-ne 2988
This theorem is referenced by:  necon4abid  3027  blssioo  23400  metdstri  23456  rrxmvallem  24008  dchrpt  25851  lgsquad3  25971  eupth2lem2  28004  lkrpssN  36456  dochshpsat  38747
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