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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, pm3.14 743, 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 832 | . . 3 ⊢ (DECID 𝜑 → DECID ¬ 𝜑) | |
2 | dcn 832 | . . 3 ⊢ (DECID 𝜓 → DECID ¬ 𝜓) | |
3 | dcor 924 | . . 3 ⊢ (DECID ¬ 𝜑 → (DECID ¬ 𝜓 → DECID (¬ 𝜑 ∨ ¬ 𝜓))) | |
4 | 1, 2, 3 | syl2im 38 | . 2 ⊢ (DECID 𝜑 → (DECID 𝜓 → DECID (¬ 𝜑 ∨ ¬ 𝜓))) |
5 | pm3.11dc 946 | . 2 ⊢ (DECID 𝜑 → (DECID 𝜓 → (¬ (¬ 𝜑 ∨ ¬ 𝜓) → (𝜑 ∧ 𝜓)))) | |
6 | con1dc 846 | . 2 ⊢ (DECID (¬ 𝜑 ∨ ¬ 𝜓) → ((¬ (¬ 𝜑 ∨ ¬ 𝜓) → (𝜑 ∧ 𝜓)) → (¬ (𝜑 ∧ 𝜓) → (¬ 𝜑 ∨ ¬ 𝜓)))) | |
7 | 4, 5, 6 | syl6c 66 | 1 ⊢ (DECID 𝜑 → (DECID 𝜓 → (¬ (𝜑 ∧ 𝜓) → (¬ 𝜑 ∨ ¬ 𝜓)))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∨ wo 698 DECID wdc 824 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 |
This theorem depends on definitions: df-bi 116 df-stab 821 df-dc 825 |
This theorem is referenced by: (None) |
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