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Theorem pm5.71dc 867
 Description: Decidable proposition version of theorem *5.71 of [WhiteheadRussell] p. 125. (Contributed by Roy F. Longton, 23-Jun-2005.) (Modified for decidability by Jim Kingdon, 19-Apr-2018.)
Assertion
Ref Expression
pm5.71dc (DECID ψ → ((ψ → ¬ χ) → (((φ ψ) χ) ↔ (φ χ))))

Proof of Theorem pm5.71dc
StepHypRef Expression
1 orel2 644 . . . . 5 ψ → ((φ ψ) → φ))
2 orc 632 . . . . 5 (φ → (φ ψ))
31, 2impbid1 130 . . . 4 ψ → ((φ ψ) ↔ φ))
43anbi1d 438 . . 3 ψ → (((φ ψ) χ) ↔ (φ χ)))
54a1i 9 . 2 (DECID ψ → (¬ ψ → (((φ ψ) χ) ↔ (φ χ))))
6 pm2.21 547 . . 3 χ → (χ → ((φ ψ) ↔ φ)))
76pm5.32rd 424 . 2 χ → (((φ ψ) χ) ↔ (φ χ)))
85, 7jadc 759 1 (DECID ψ → ((ψ → ¬ χ) → (((φ ψ) χ) ↔ (φ χ))))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 628  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: (None)
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