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Theorem con4biddc 754
 Description: A contraposition deduction. (Contributed by Jim Kingdon, 18-May-2018.)
Hypothesis
Ref Expression
con4biddc.1 (𝜑 → (DECID 𝜓 → (DECID 𝜒 → (¬ 𝜓 ↔ ¬ 𝜒))))
Assertion
Ref Expression
con4biddc (𝜑 → (DECID 𝜓 → (DECID 𝜒 → (𝜓𝜒))))

Proof of Theorem con4biddc
StepHypRef Expression
1 con4biddc.1 . . . . . 6 (𝜑 → (DECID 𝜓 → (DECID 𝜒 → (¬ 𝜓 ↔ ¬ 𝜒))))
2 bi2 121 . . . . . 6 ((¬ 𝜓 ↔ ¬ 𝜒) → (¬ 𝜒 → ¬ 𝜓))
31, 2syl8 65 . . . . 5 (𝜑 → (DECID 𝜓 → (DECID 𝜒 → (¬ 𝜒 → ¬ 𝜓))))
4 condc 749 . . . . . 6 (DECID 𝜒 → ((¬ 𝜒 → ¬ 𝜓) → (𝜓𝜒)))
54a2i 11 . . . . 5 ((DECID 𝜒 → (¬ 𝜒 → ¬ 𝜓)) → (DECID 𝜒 → (𝜓𝜒)))
63, 5syl6 29 . . . 4 (𝜑 → (DECID 𝜓 → (DECID 𝜒 → (𝜓𝜒))))
76imp31 243 . . 3 (((𝜑DECID 𝜓) ∧ DECID 𝜒) → (𝜓𝜒))
8 bi1 111 . . . . . 6 ((¬ 𝜓 ↔ ¬ 𝜒) → (¬ 𝜓 → ¬ 𝜒))
91, 8syl8 65 . . . . 5 (𝜑 → (DECID 𝜓 → (DECID 𝜒 → (¬ 𝜓 → ¬ 𝜒))))
10 condc 749 . . . . . 6 (DECID 𝜓 → ((¬ 𝜓 → ¬ 𝜒) → (𝜒𝜓)))
1110imim2d 48 . . . . 5 (DECID 𝜓 → ((DECID 𝜒 → (¬ 𝜓 → ¬ 𝜒)) → (DECID 𝜒 → (𝜒𝜓))))
129, 11sylcom 25 . . . 4 (𝜑 → (DECID 𝜓 → (DECID 𝜒 → (𝜒𝜓))))
1312imp31 243 . . 3 (((𝜑DECID 𝜓) ∧ DECID 𝜒) → (𝜒𝜓))
147, 13impbid 120 . 2 (((𝜑DECID 𝜓) ∧ DECID 𝜒) → (𝜓𝜒))
1514exp31 346 1 (𝜑 → (DECID 𝜓 → (DECID 𝜒 → (𝜓𝜒))))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ 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-in2 545  ax-io 630 This theorem depends on definitions:  df-bi 110  df-dc 743 This theorem is referenced by:  necon4abiddc  2278
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