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Mirrors > Home > ILE Home > Th. List > ianordc | GIF version |
Description: Negated conjunction in terms of disjunction (DeMorgan's law). Theorem *4.51 of [WhiteheadRussell] p. 120, but where one proposition is decidable. The reverse direction, pm3.14 669, holds for all propositions, but the equivalence only holds where one proposition is decidable. (Contributed by Jim Kingdon, 21-Apr-2018.) |
Ref | Expression |
---|---|
ianordc | ⊢ (DECID φ → (¬ (φ ∧ ψ) ↔ (¬ φ ∨ ¬ ψ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imnan 623 | . 2 ⊢ ((φ → ¬ ψ) ↔ ¬ (φ ∧ ψ)) | |
2 | pm4.62dc 797 | . 2 ⊢ (DECID φ → ((φ → ¬ ψ) ↔ (¬ φ ∨ ¬ ψ))) | |
3 | 1, 2 | syl5bbr 183 | 1 ⊢ (DECID φ → (¬ (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: anordc 862 19.33bdc 1518 nn0n0n1ge2b 8096 |
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