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Theorem orandc 883
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 pm4.56 842 . 2 ((¬ 𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑𝜓))
2 dcn 782 . . . . 5 (DECID 𝜑DECID ¬ 𝜑)
32adantr 270 . . . 4 ((DECID 𝜑DECID 𝜓) → DECID ¬ 𝜑)
4 dcn 782 . . . . 5 (DECID 𝜓DECID ¬ 𝜓)
54adantl 271 . . . 4 ((DECID 𝜑DECID 𝜓) → DECID ¬ 𝜓)
6 dcan 878 . . . 4 (DECID ¬ 𝜑 → (DECID ¬ 𝜓DECID𝜑 ∧ ¬ 𝜓)))
73, 5, 6sylc 61 . . 3 ((DECID 𝜑DECID 𝜓) → DECID𝜑 ∧ ¬ 𝜓))
8 dcor 879 . . . 4 (DECID 𝜑 → (DECID 𝜓DECID (𝜑𝜓)))
98imp 122 . . 3 ((DECID 𝜑DECID 𝜓) → DECID (𝜑𝜓))
10 con2bidc 805 . . 3 (DECID𝜑 ∧ ¬ 𝜓) → (DECID (𝜑𝜓) → (((¬ 𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑𝜓)) ↔ ((𝜑𝜓) ↔ ¬ (¬ 𝜑 ∧ ¬ 𝜓)))))
117, 9, 10sylc 61 . 2 ((DECID 𝜑DECID 𝜓) → (((¬ 𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑𝜓)) ↔ ((𝜑𝜓) ↔ ¬ (¬ 𝜑 ∧ ¬ 𝜓))))
121, 11mpbii 146 1 ((DECID 𝜑DECID 𝜓) → ((𝜑𝜓) ↔ ¬ (¬ 𝜑 ∧ ¬ 𝜓)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wo 662  DECID wdc 778
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 779
This theorem is referenced by:  gcdaddm  10769
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