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Theorem pm4.54dc 839
Description: Theorem *4.54 of [WhiteheadRussell] p. 120, for decidable propositions. One form of DeMorgan's law. (Contributed by Jim Kingdon, 2-May-2018.)
Assertion
Ref Expression
pm4.54dc (DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑𝜓) ↔ ¬ (𝜑 ∨ ¬ 𝜓))))

Proof of Theorem pm4.54dc
StepHypRef Expression
1 dcn 780 . . . . 5 (DECID 𝜑DECID ¬ 𝜑)
2 dfandc 812 . . . . 5 (DECID ¬ 𝜑 → (DECID 𝜓 → ((¬ 𝜑𝜓) ↔ ¬ (¬ 𝜑 → ¬ 𝜓))))
31, 2syl 14 . . . 4 (DECID 𝜑 → (DECID 𝜓 → ((¬ 𝜑𝜓) ↔ ¬ (¬ 𝜑 → ¬ 𝜓))))
43imp 122 . . 3 ((DECID 𝜑DECID 𝜓) → ((¬ 𝜑𝜓) ↔ ¬ (¬ 𝜑 → ¬ 𝜓)))
5 pm4.66dc 836 . . . . 5 (DECID 𝜑 → ((¬ 𝜑 → ¬ 𝜓) ↔ (𝜑 ∨ ¬ 𝜓)))
65adantr 270 . . . 4 ((DECID 𝜑DECID 𝜓) → ((¬ 𝜑 → ¬ 𝜓) ↔ (𝜑 ∨ ¬ 𝜓)))
76notbid 625 . . 3 ((DECID 𝜑DECID 𝜓) → (¬ (¬ 𝜑 → ¬ 𝜓) ↔ ¬ (𝜑 ∨ ¬ 𝜓)))
84, 7bitrd 186 . 2 ((DECID 𝜑DECID 𝜓) → ((¬ 𝜑𝜓) ↔ ¬ (𝜑 ∨ ¬ 𝜓)))
98ex 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:  pm4.55dc  880
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