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Theorem necon1abii 2871
Description: Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)
Hypothesis
Ref Expression
necon1abii.1 𝜑𝐴 = 𝐵)
Assertion
Ref Expression
necon1abii (𝐴𝐵𝜑)

Proof of Theorem necon1abii
StepHypRef Expression
1 notnotb 304 . 2 (𝜑 ↔ ¬ ¬ 𝜑)
2 necon1abii.1 . . 3 𝜑𝐴 = 𝐵)
32necon3bbii 2870 . 2 (¬ ¬ 𝜑𝐴𝐵)
41, 3bitr2i 265 1 (𝐴𝐵𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196   = wceq 1523  wne 2823
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 2824
This theorem is referenced by:  necon2abii  2873  marypha1lem  8380  npomex  9856  uniinn0  29492
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