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Theorem anordc 862
Description: Conjunction in terms of disjunction (DeMorgan's law). Theorem *4.5 of [WhiteheadRussell] p. 120, but where the propositions are decidable. The forward direction, pm3.1 670, holds for all propositions, but the equivalence only holds given decidability. (Contributed by Jim Kingdon, 21-Apr-2018.)
Assertion
Ref Expression
anordc (DECID φ → (DECID ψ → ((φ ψ) ↔ ¬ (¬ φ ¬ ψ))))

Proof of Theorem anordc
StepHypRef Expression
1 dcan 841 . 2 (DECID φ → (DECID ψDECID (φ ψ)))
2 ianordc 798 . . . . 5 (DECID φ → (¬ (φ ψ) ↔ (¬ φ ¬ ψ)))
32bicomd 129 . . . 4 (DECID φ → ((¬ φ ¬ ψ) ↔ ¬ (φ ψ)))
43a1d 22 . . 3 (DECID φ → (DECID (φ ψ) → ((¬ φ ¬ ψ) ↔ ¬ (φ ψ))))
54con2biddc 773 . 2 (DECID φ → (DECID (φ ψ) → ((φ ψ) ↔ ¬ (¬ φ ¬ ψ))))
61, 5syld 40 1 (DECID φ → (DECID ψ → ((φ ψ) ↔ ¬ (¬ φ ¬ ψ))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98   wo 628  DECID wdc 741
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 629
This theorem depends on definitions:  df-bi 110  df-dc 742
This theorem is referenced by:  pm3.11dc  863  dn1dc  866
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