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Theorem necon2bbiddc 2266
Description: Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
Hypothesis
Ref Expression
necon2bbiddc.1 (φ → (DECID A = B → (ψAB)))
Assertion
Ref Expression
necon2bbiddc (φ → (DECID A = B → (A = B ↔ ¬ ψ)))

Proof of Theorem necon2bbiddc
StepHypRef Expression
1 necon2bbiddc.1 . . . 4 (φ → (DECID A = B → (ψAB)))
2 bicom 128 . . . 4 ((ψAB) ↔ (ABψ))
31, 2syl6ib 150 . . 3 (φ → (DECID A = B → (ABψ)))
43necon1bbiddc 2262 . 2 (φ → (DECID A = B → (¬ ψA = B)))
5 bicom 128 . 2 ((¬ ψA = B) ↔ (A = B ↔ ¬ ψ))
64, 5syl6ib 150 1 (φ → (DECID A = B → (A = B ↔ ¬ ψ)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 98  DECID wdc 741   = wceq 1242  wne 2201
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 544  ax-in2 545  ax-io 629
This theorem depends on definitions:  df-bi 110  df-dc 742  df-ne 2203
This theorem is referenced by: (None)
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