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Theorem necon1bidc 2272
Description: Contrapositive inference for inequality. (Contributed by Jim Kingdon, 15-May-2018.)
Hypothesis
Ref Expression
necon1bidc.1 (DECID 𝐴 = 𝐵 → (𝐴𝐵𝜑))
Assertion
Ref Expression
necon1bidc (DECID 𝐴 = 𝐵 → (¬ 𝜑𝐴 = 𝐵))

Proof of Theorem necon1bidc
StepHypRef Expression
1 df-ne 2221 . . 3 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
2 necon1bidc.1 . . 3 (DECID 𝐴 = 𝐵 → (𝐴𝐵𝜑))
31, 2syl5bir 146 . 2 (DECID 𝐴 = 𝐵 → (¬ 𝐴 = 𝐵𝜑))
4 con1dc 764 . 2 (DECID 𝐴 = 𝐵 → ((¬ 𝐴 = 𝐵𝜑) → (¬ 𝜑𝐴 = 𝐵)))
53, 4mpd 13 1 (DECID 𝐴 = 𝐵 → (¬ 𝜑𝐴 = 𝐵))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  DECID wdc 753   = wceq 1259  wne 2220
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640
This theorem depends on definitions:  df-bi 114  df-dc 754  df-ne 2221
This theorem is referenced by: (None)
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