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Theorem con4biddc 753
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 748 . . . . . 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 (((φ DECID 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 748 . . . . . 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 (((φ DECID ψ) DECID χ) → (χψ))
147, 13impbid 120 . 2 (((φ DECID 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 741
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 629
This theorem depends on definitions:  df-bi 110  df-dc 742
This theorem is referenced by:  necon4abiddc  2272
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