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Theorem con2biddc 774
 Description: A contraposition deduction. (Contributed by Jim Kingdon, 11-Apr-2018.)
Hypothesis
Ref Expression
con2biddc.1 (𝜑 → (DECID 𝜒 → (𝜓 ↔ ¬ 𝜒)))
Assertion
Ref Expression
con2biddc (𝜑 → (DECID 𝜒 → (𝜒 ↔ ¬ 𝜓)))

Proof of Theorem con2biddc
StepHypRef Expression
1 con2biddc.1 . . . 4 (𝜑 → (DECID 𝜒 → (𝜓 ↔ ¬ 𝜒)))
2 bicom 128 . . . 4 ((𝜓 ↔ ¬ 𝜒) ↔ (¬ 𝜒𝜓))
31, 2syl6ib 150 . . 3 (𝜑 → (DECID 𝜒 → (¬ 𝜒𝜓)))
43con1biddc 770 . 2 (𝜑 → (DECID 𝜒 → (¬ 𝜓𝜒)))
5 bicom 128 . 2 ((¬ 𝜓𝜒) ↔ (𝜒 ↔ ¬ 𝜓))
64, 5syl6ib 150 1 (𝜑 → (DECID 𝜒 → (𝜒 ↔ ¬ 𝜓)))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 98  DECID wdc 742 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 630 This theorem depends on definitions:  df-bi 110  df-dc 743 This theorem is referenced by:  anordc  863  xor3dc  1278
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