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Theorem orandc 924
 Description: Disjunction in terms of conjunction (De Morgan's law), for decidable propositions. Compare Theorem *4.57 of [WhiteheadRussell] p. 120. (Contributed by Jim Kingdon, 13-Dec-2021.)
Assertion
Ref Expression
orandc ((DECID 𝜑DECID 𝜓) → ((𝜑𝜓) ↔ ¬ (¬ 𝜑 ∧ ¬ 𝜓)))

Proof of Theorem orandc
StepHypRef Expression
1 pm4.56 770 . 2 ((¬ 𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑𝜓))
2 dcn 828 . . . . 5 (DECID 𝜑DECID ¬ 𝜑)
32adantr 274 . . . 4 ((DECID 𝜑DECID 𝜓) → DECID ¬ 𝜑)
4 dcn 828 . . . . 5 (DECID 𝜓DECID ¬ 𝜓)
54adantl 275 . . . 4 ((DECID 𝜑DECID 𝜓) → DECID ¬ 𝜓)
6 dcan 919 . . . 4 (DECID ¬ 𝜑 → (DECID ¬ 𝜓DECID𝜑 ∧ ¬ 𝜓)))
73, 5, 6sylc 62 . . 3 ((DECID 𝜑DECID 𝜓) → DECID𝜑 ∧ ¬ 𝜓))
8 dcor 920 . . . 4 (DECID 𝜑 → (DECID 𝜓DECID (𝜑𝜓)))
98imp 123 . . 3 ((DECID 𝜑DECID 𝜓) → DECID (𝜑𝜓))
10 con2bidc 861 . . 3 (DECID𝜑 ∧ ¬ 𝜓) → (DECID (𝜑𝜓) → (((¬ 𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑𝜓)) ↔ ((𝜑𝜓) ↔ ¬ (¬ 𝜑 ∧ ¬ 𝜓)))))
117, 9, 10sylc 62 . 2 ((DECID 𝜑DECID 𝜓) → (((¬ 𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑𝜓)) ↔ ((𝜑𝜓) ↔ ¬ (¬ 𝜑 ∧ ¬ 𝜓))))
121, 11mpbii 147 1 ((DECID 𝜑DECID 𝜓) → ((𝜑𝜓) ↔ ¬ (¬ 𝜑 ∧ ¬ 𝜓)))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 698  DECID wdc 820 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699 This theorem depends on definitions:  df-bi 116  df-stab 817  df-dc 821 This theorem is referenced by:  gcdaddm  11706
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