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Theorem necon2abii 3066
Description: Contrapositive inference for inequality. (Contributed by NM, 2-Mar-2007.)
Hypothesis
Ref Expression
necon2abii.1 (𝐴 = 𝐵 ↔ ¬ 𝜑)
Assertion
Ref Expression
necon2abii (𝜑𝐴𝐵)

Proof of Theorem necon2abii
StepHypRef Expression
1 necon2abii.1 . . . 4 (𝐴 = 𝐵 ↔ ¬ 𝜑)
21bicomi 225 . . 3 𝜑𝐴 = 𝐵)
32necon1abii 3064 . 2 (𝐴𝐵𝜑)
43bicomi 225 1 (𝜑𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 207   = wceq 1528  wne 3016
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 208  df-ne 3017
This theorem is referenced by:  locfindis  22068  flimsncls  22524  tsmsgsum  22676  wilthlem2  25574  topdifinffinlem  34511  ismblfin  34815  elnev  40650
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