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Theorem necon2bbii 3058
Description: Contrapositive inference for inequality. (Contributed by NM, 13-Apr-2007.)
Hypothesis
Ref Expression
necon2bbii.1 (𝜑𝐴𝐵)
Assertion
Ref Expression
necon2bbii (𝐴 = 𝐵 ↔ ¬ 𝜑)

Proof of Theorem necon2bbii
StepHypRef Expression
1 necon2bbii.1 . . . 4 (𝜑𝐴𝐵)
21bicomi 227 . . 3 (𝐴𝐵𝜑)
32necon1bbii 3056 . 2 𝜑𝐴 = 𝐵)
43bicomi 227 1 (𝐴 = 𝐵 ↔ ¬ 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 209   = wceq 1538  wne 3007
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 3008
This theorem is referenced by:  xpeq0  5990  dmsn0  6039  disjex  30329  disjexc  30330  suppss3  30447
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