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Theorem necon1abiddc 2261
Description: Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 16-May-2018.)
Hypothesis
Ref Expression
necon1abiddc.1 (φ → (DECID ψ → (¬ ψA = B)))
Assertion
Ref Expression
necon1abiddc (φ → (DECID ψ → (ABψ)))

Proof of Theorem necon1abiddc
StepHypRef Expression
1 necon1abiddc.1 . . 3 (φ → (DECID ψ → (¬ ψA = B)))
21con1biddc 769 . 2 (φ → (DECID ψ → (¬ A = Bψ)))
3 df-ne 2203 . . 3 (AB ↔ ¬ A = B)
43bibi1i 217 . 2 ((ABψ) ↔ (¬ A = Bψ))
52, 4syl6ibr 151 1 (φ → (DECID ψ → (ABψ)))
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:  necon2abiddc  2265
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