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Theorem necon1bidc 2231
Description: Contrapositive inference for inequality. (Contributed by Jim Kingdon, 15-May-2018.)
Hypothesis
Ref Expression
necon1bidc.1 (DECID A = B → (ABφ))
Assertion
Ref Expression
necon1bidc (DECID A = B → (¬ φA = B))

Proof of Theorem necon1bidc
StepHypRef Expression
1 df-ne 2184 . . 3 (AB ↔ ¬ A = B)
2 necon1bidc.1 . . 3 (DECID A = B → (ABφ))
31, 2syl5bir 142 . 2 (DECID A = B → (¬ A = Bφ))
4 con1dc 741 . 2 (DECID A = B → ((¬ A = Bφ) → (¬ φA = B)))
53, 4mpd 13 1 (DECID A = B → (¬ φA = B))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  DECID wdc 730   = wceq 1226  wne 2182
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617
This theorem depends on definitions:  df-bi 110  df-dc 731  df-ne 2184
This theorem is referenced by: (None)
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