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Theorem pm5.71dc 879
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) → (((𝜑𝜓) ∧ 𝜒) ↔ (𝜑𝜒))))

Proof of Theorem pm5.71dc
StepHypRef Expression
1 orel2 655 . . . . 5 𝜓 → ((𝜑𝜓) → 𝜑))
2 orc 643 . . . . 5 (𝜑 → (𝜑𝜓))
31, 2impbid1 134 . . . 4 𝜓 → ((𝜑𝜓) ↔ 𝜑))
43anbi1d 446 . . 3 𝜓 → (((𝜑𝜓) ∧ 𝜒) ↔ (𝜑𝜒)))
54a1i 9 . 2 (DECID 𝜓 → (¬ 𝜓 → (((𝜑𝜓) ∧ 𝜒) ↔ (𝜑𝜒))))
6 pm2.21 557 . . 3 𝜒 → (𝜒 → ((𝜑𝜓) ↔ 𝜑)))
76pm5.32rd 432 . 2 𝜒 → (((𝜑𝜓) ∧ 𝜒) ↔ (𝜑𝜒)))
85, 7jadc 771 1 (DECID 𝜓 → ((𝜓 → ¬ 𝜒) → (((𝜑𝜓) ∧ 𝜒) ↔ (𝜑𝜒))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  wb 102  wo 639  DECID wdc 753
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in2 555  ax-io 640
This theorem depends on definitions:  df-bi 114  df-dc 754
This theorem is referenced by: (None)
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