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Theorem necon1bddc 2332
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 2256 . . . 4 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
32imbi1i 236 . . 3 ((𝐴𝐵𝜓) ↔ (¬ 𝐴 = 𝐵𝜓))
41, 3syl6ib 159 . 2 (𝜑 → (DECID 𝐴 = 𝐵 → (¬ 𝐴 = 𝐵𝜓)))
5 con1dc 791 . 2 (DECID 𝐴 = 𝐵 → ((¬ 𝐴 = 𝐵𝜓) → (¬ 𝜓𝐴 = 𝐵)))
64, 5sylcom 28 1 (𝜑 → (DECID 𝐴 = 𝐵 → (¬ 𝜓𝐴 = 𝐵)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  DECID wdc 780   = wceq 1289  wne 2255
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665
This theorem depends on definitions:  df-bi 115  df-dc 781  df-ne 2256
This theorem is referenced by:  necon1ddc  2333
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