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Theorem necon4bddc 2354
Description: Contrapositive inference for inequality. (Contributed by Jim Kingdon, 17-May-2018.)
Hypothesis
Ref Expression
necon4bddc.1 (𝜑 → (DECID 𝜓 → (¬ 𝜓𝐴𝐵)))
Assertion
Ref Expression
necon4bddc (𝜑 → (DECID 𝜓 → (𝐴 = 𝐵𝜓)))

Proof of Theorem necon4bddc
StepHypRef Expression
1 necon4bddc.1 . . 3 (𝜑 → (DECID 𝜓 → (¬ 𝜓𝐴𝐵)))
2 df-ne 2284 . . 3 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
31, 2syl8ib 165 . 2 (𝜑 → (DECID 𝜓 → (¬ 𝜓 → ¬ 𝐴 = 𝐵)))
4 condc 821 . 2 (DECID 𝜓 → ((¬ 𝜓 → ¬ 𝐴 = 𝐵) → (𝐴 = 𝐵𝜓)))
53, 4sylcom 28 1 (𝜑 → (DECID 𝜓 → (𝐴 = 𝐵𝜓)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  DECID wdc 802   = wceq 1314  wne 2283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681
This theorem depends on definitions:  df-bi 116  df-stab 799  df-dc 803  df-ne 2284
This theorem is referenced by: (None)
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