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Theorem pm4.55dc 880
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 839 . . . . 5 (DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑𝜓) ↔ ¬ (𝜑 ∨ ¬ 𝜓))))
21imp 122 . . . 4 ((DECID 𝜑DECID 𝜓) → ((¬ 𝜑𝜓) ↔ ¬ (𝜑 ∨ ¬ 𝜓)))
3 dcn 780 . . . . . . . . 9 (DECID 𝜓DECID ¬ 𝜓)
43anim2i 334 . . . . . . . 8 ((DECID 𝜑DECID 𝜓) → (DECID 𝜑DECID ¬ 𝜓))
5 dcor 877 . . . . . . . . 9 (DECID 𝜑 → (DECID ¬ 𝜓DECID (𝜑 ∨ ¬ 𝜓)))
65imp 122 . . . . . . . 8 ((DECID 𝜑DECID ¬ 𝜓) → DECID (𝜑 ∨ ¬ 𝜓))
74, 6syl 14 . . . . . . 7 ((DECID 𝜑DECID 𝜓) → DECID (𝜑 ∨ ¬ 𝜓))
8 dcn 780 . . . . . . . . 9 (DECID 𝜑DECID ¬ 𝜑)
9 dcan 876 . . . . . . . . 9 (DECID ¬ 𝜑 → (DECID 𝜓DECID𝜑𝜓)))
108, 9syl 14 . . . . . . . 8 (DECID 𝜑 → (DECID 𝜓DECID𝜑𝜓)))
1110imp 122 . . . . . . 7 ((DECID 𝜑DECID 𝜓) → DECID𝜑𝜓))
127, 11jca 300 . . . . . 6 ((DECID 𝜑DECID 𝜓) → (DECID (𝜑 ∨ ¬ 𝜓) ∧ DECID𝜑𝜓)))
13 con2bidc 803 . . . . . . 7 (DECID (𝜑 ∨ ¬ 𝜓) → (DECID𝜑𝜓) → (((𝜑 ∨ ¬ 𝜓) ↔ ¬ (¬ 𝜑𝜓)) ↔ ((¬ 𝜑𝜓) ↔ ¬ (𝜑 ∨ ¬ 𝜓)))))
1413imp 122 . . . . . 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 156 . . . 4 ((DECID 𝜑DECID 𝜓) → (((¬ 𝜑𝜓) ↔ ¬ (𝜑 ∨ ¬ 𝜓)) → ((𝜑 ∨ ¬ 𝜓) ↔ ¬ (¬ 𝜑𝜓))))
172, 16mpd 13 . . 3 ((DECID 𝜑DECID 𝜓) → ((𝜑 ∨ ¬ 𝜓) ↔ ¬ (¬ 𝜑𝜓)))
1817bicomd 139 . 2 ((DECID 𝜑DECID 𝜓) → (¬ (¬ 𝜑𝜓) ↔ (𝜑 ∨ ¬ 𝜓)))
1918ex 113 1 (DECID 𝜑 → (DECID 𝜓 → (¬ (¬ 𝜑𝜓) ↔ (𝜑 ∨ ¬ 𝜓))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wo 662  DECID wdc 776
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663
This theorem depends on definitions:  df-bi 115  df-dc 777
This theorem is referenced by: (None)
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