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Theorem necon1bddc 2276
Description: Contrapositive deduction for inequality. (Contributed by Jim Kingdon, 19-May-2018.)
Hypothesis
Ref Expression
necon1bddc.1 (φ → (DECID A = B → (ABψ)))
Assertion
Ref Expression
necon1bddc (φ → (DECID A = B → (¬ ψA = B)))

Proof of Theorem necon1bddc
StepHypRef Expression
1 necon1bddc.1 . . 3 (φ → (DECID A = B → (ABψ)))
2 df-ne 2203 . . . 4 (AB ↔ ¬ A = B)
32imbi1i 227 . . 3 ((ABψ) ↔ (¬ A = Bψ))
41, 3syl6ib 150 . 2 (φ → (DECID A = B → (¬ A = Bψ)))
5 con1dc 752 . 2 (DECID A = B → ((¬ A = Bψ) → (¬ ψA = B)))
64, 5sylcom 25 1 (φ → (DECID A = B → (¬ ψA = B)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  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:  necon1ddc  2277
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