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Mirrors > Home > ILE Home > Th. List > pm3.13dc | GIF version |
Description: Theorem *3.13 of [WhiteheadRussell] p. 111, but for decidable propositions. The converse, pm3.14 669, holds for all propositions. (Contributed by Jim Kingdon, 22-Apr-2018.) |
Ref | Expression |
---|---|
pm3.13dc | ⊢ (DECID φ → (DECID ψ → (¬ (φ ∧ ψ) → (¬ φ ∨ ¬ ψ)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dcn 745 | . . 3 ⊢ (DECID φ → DECID ¬ φ) | |
2 | dcn 745 | . . 3 ⊢ (DECID ψ → DECID ¬ ψ) | |
3 | dcor 842 | . . 3 ⊢ (DECID ¬ φ → (DECID ¬ ψ → DECID (¬ φ ∨ ¬ ψ))) | |
4 | 1, 2, 3 | syl2im 34 | . 2 ⊢ (DECID φ → (DECID ψ → DECID (¬ φ ∨ ¬ ψ))) |
5 | pm3.11dc 863 | . 2 ⊢ (DECID φ → (DECID ψ → (¬ (¬ φ ∨ ¬ ψ) → (φ ∧ ψ)))) | |
6 | con1dc 752 | . 2 ⊢ (DECID (¬ φ ∨ ¬ ψ) → ((¬ (¬ φ ∨ ¬ ψ) → (φ ∧ ψ)) → (¬ (φ ∧ ψ) → (¬ φ ∨ ¬ ψ)))) | |
7 | 4, 5, 6 | syl6c 60 | 1 ⊢ (DECID φ → (DECID ψ → (¬ (φ ∧ ψ) → (¬ φ ∨ ¬ ψ)))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 97 ∨ 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: (None) |
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