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Theorem necon4aidc 2251
Description: Contrapositive inference for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
Hypothesis
Ref Expression
necon4aidc.1 (DECID A = B → (AB → ¬ φ))
Assertion
Ref Expression
necon4aidc (DECID A = B → (φA = B))

Proof of Theorem necon4aidc
StepHypRef Expression
1 df-ne 2188 . . 3 (AB ↔ ¬ A = B)
2 necon4aidc.1 . . 3 (DECID A = B → (AB → ¬ φ))
31, 2syl5bir 142 . 2 (DECID A = B → (¬ A = B → ¬ φ))
4 condc 740 . 2 (DECID A = B → ((¬ A = B → ¬ φ) → (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 733   = wceq 1228  wne 2186
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-in2 533  ax-io 617
This theorem depends on definitions:  df-bi 110  df-dc 734  df-ne 2188
This theorem is referenced by:  necon4idc  2252
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