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Theorem pm4.55dc 846
Description: Theorem *4.55 of [WhiteheadRussell] p. 120, for decidable propositions. (Contributed by Jim Kingdon, 2-May-2018.)
Assertion
Ref Expression
pm4.55dc (DECID 𝜑 → (DECID 𝜓 → (¬ (¬ 𝜑𝜓) ↔ (𝜑 ∨ ¬ 𝜓))))

Proof of Theorem pm4.55dc
StepHypRef Expression
1 pm4.54dc 805 . . . . 5 (DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑𝜓) ↔ ¬ (𝜑 ∨ ¬ 𝜓))))
21imp 115 . . . 4 ((DECID 𝜑DECID 𝜓) → ((¬ 𝜑𝜓) ↔ ¬ (𝜑 ∨ ¬ 𝜓)))
3 dcn 746 . . . . . . . . 9 (DECID 𝜓DECID ¬ 𝜓)
43anim2i 324 . . . . . . . 8 ((DECID 𝜑DECID 𝜓) → (DECID 𝜑DECID ¬ 𝜓))
5 dcor 843 . . . . . . . . 9 (DECID 𝜑 → (DECID ¬ 𝜓DECID (𝜑 ∨ ¬ 𝜓)))
65imp 115 . . . . . . . 8 ((DECID 𝜑DECID ¬ 𝜓) → DECID (𝜑 ∨ ¬ 𝜓))
74, 6syl 14 . . . . . . 7 ((DECID 𝜑DECID 𝜓) → DECID (𝜑 ∨ ¬ 𝜓))
8 dcn 746 . . . . . . . . 9 (DECID 𝜑DECID ¬ 𝜑)
9 dcan 842 . . . . . . . . 9 (DECID ¬ 𝜑 → (DECID 𝜓DECID𝜑𝜓)))
108, 9syl 14 . . . . . . . 8 (DECID 𝜑 → (DECID 𝜓DECID𝜑𝜓)))
1110imp 115 . . . . . . 7 ((DECID 𝜑DECID 𝜓) → DECID𝜑𝜓))
127, 11jca 290 . . . . . 6 ((DECID 𝜑DECID 𝜓) → (DECID (𝜑 ∨ ¬ 𝜓) ∧ DECID𝜑𝜓)))
13 con2bidc 769 . . . . . . 7 (DECID (𝜑 ∨ ¬ 𝜓) → (DECID𝜑𝜓) → (((𝜑 ∨ ¬ 𝜓) ↔ ¬ (¬ 𝜑𝜓)) ↔ ((¬ 𝜑𝜓) ↔ ¬ (𝜑 ∨ ¬ 𝜓)))))
1413imp 115 . . . . . 6 ((DECID (𝜑 ∨ ¬ 𝜓) ∧ DECID𝜑𝜓)) → (((𝜑 ∨ ¬ 𝜓) ↔ ¬ (¬ 𝜑𝜓)) ↔ ((¬ 𝜑𝜓) ↔ ¬ (𝜑 ∨ ¬ 𝜓))))
1512, 14syl 14 . . . . 5 ((DECID 𝜑DECID 𝜓) → (((𝜑 ∨ ¬ 𝜓) ↔ ¬ (¬ 𝜑𝜓)) ↔ ((¬ 𝜑𝜓) ↔ ¬ (𝜑 ∨ ¬ 𝜓))))
1615biimprd 147 . . . 4 ((DECID 𝜑DECID 𝜓) → (((¬ 𝜑𝜓) ↔ ¬ (𝜑 ∨ ¬ 𝜓)) → ((𝜑 ∨ ¬ 𝜓) ↔ ¬ (¬ 𝜑𝜓))))
172, 16mpd 13 . . 3 ((DECID 𝜑DECID 𝜓) → ((𝜑 ∨ ¬ 𝜓) ↔ ¬ (¬ 𝜑𝜓)))
1817bicomd 129 . 2 ((DECID 𝜑DECID 𝜓) → (¬ (¬ 𝜑𝜓) ↔ (𝜑 ∨ ¬ 𝜓)))
1918ex 108 1 (DECID 𝜑 → (DECID 𝜓 → (¬ (¬ 𝜑𝜓) ↔ (𝜑 ∨ ¬ 𝜓))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 97  wb 98  wo 629  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-in1 544  ax-in2 545  ax-io 630
This theorem depends on definitions:  df-bi 110  df-dc 743
This theorem is referenced by: (None)
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