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Theorem con4biddc 825
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 129 . . . . . 6 ((¬ 𝜓 ↔ ¬ 𝜒) → (¬ 𝜒 → ¬ 𝜓))
31, 2syl8 71 . . . . 5 (𝜑 → (DECID 𝜓 → (DECID 𝜒 → (¬ 𝜒 → ¬ 𝜓))))
4 condc 821 . . . . . 6 (DECID 𝜒 → ((¬ 𝜒 → ¬ 𝜓) → (𝜓𝜒)))
54a2i 11 . . . . 5 ((DECID 𝜒 → (¬ 𝜒 → ¬ 𝜓)) → (DECID 𝜒 → (𝜓𝜒)))
63, 5syl6 33 . . . 4 (𝜑 → (DECID 𝜓 → (DECID 𝜒 → (𝜓𝜒))))
76imp31 254 . . 3 (((𝜑DECID 𝜓) ∧ DECID 𝜒) → (𝜓𝜒))
8 bi1 117 . . . . . 6 ((¬ 𝜓 ↔ ¬ 𝜒) → (¬ 𝜓 → ¬ 𝜒))
91, 8syl8 71 . . . . 5 (𝜑 → (DECID 𝜓 → (DECID 𝜒 → (¬ 𝜓 → ¬ 𝜒))))
10 condc 821 . . . . . 6 (DECID 𝜓 → ((¬ 𝜓 → ¬ 𝜒) → (𝜒𝜓)))
1110imim2d 54 . . . . 5 (DECID 𝜓 → ((DECID 𝜒 → (¬ 𝜓 → ¬ 𝜒)) → (DECID 𝜒 → (𝜒𝜓))))
129, 11sylcom 28 . . . 4 (𝜑 → (DECID 𝜓 → (DECID 𝜒 → (𝜒𝜓))))
1312imp31 254 . . 3 (((𝜑DECID 𝜓) ∧ DECID 𝜒) → (𝜒𝜓))
147, 13impbid 128 . 2 (((𝜑DECID 𝜓) ∧ DECID 𝜒) → (𝜓𝜒))
1514exp31 359 1 (𝜑 → (DECID 𝜓 → (DECID 𝜒 → (𝜓𝜒))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  DECID wdc 802
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
This theorem is referenced by:  necon4abiddc  2356
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