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Mirrors > Home > ILE Home > Th. List > xordc1 | GIF version |
Description: Exclusive or implies the left proposition is decidable. (Contributed by Jim Kingdon, 12-Mar-2018.) |
Ref | Expression |
---|---|
xordc1 | ⊢ ((𝜑 ⊻ 𝜓) → DECID 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | andir 809 | . . 3 ⊢ (((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)) ↔ ((𝜑 ∧ ¬ (𝜑 ∧ 𝜓)) ∨ (𝜓 ∧ ¬ (𝜑 ∧ 𝜓)))) | |
2 | simpl 108 | . . . 4 ⊢ ((𝜑 ∧ ¬ (𝜑 ∧ 𝜓)) → 𝜑) | |
3 | imnan 680 | . . . . . 6 ⊢ ((𝜓 → ¬ 𝜑) ↔ ¬ (𝜓 ∧ 𝜑)) | |
4 | ancom 264 | . . . . . 6 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜓 ∧ 𝜑)) | |
5 | 3, 4 | xchbinxr 673 | . . . . 5 ⊢ ((𝜓 → ¬ 𝜑) ↔ ¬ (𝜑 ∧ 𝜓)) |
6 | pm3.35 345 | . . . . 5 ⊢ ((𝜓 ∧ (𝜓 → ¬ 𝜑)) → ¬ 𝜑) | |
7 | 5, 6 | sylan2br 286 | . . . 4 ⊢ ((𝜓 ∧ ¬ (𝜑 ∧ 𝜓)) → ¬ 𝜑) |
8 | 2, 7 | orim12i 749 | . . 3 ⊢ (((𝜑 ∧ ¬ (𝜑 ∧ 𝜓)) ∨ (𝜓 ∧ ¬ (𝜑 ∧ 𝜓))) → (𝜑 ∨ ¬ 𝜑)) |
9 | 1, 8 | sylbi 120 | . 2 ⊢ (((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)) → (𝜑 ∨ ¬ 𝜑)) |
10 | df-xor 1365 | . 2 ⊢ ((𝜑 ⊻ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓))) | |
11 | df-dc 825 | . 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑)) | |
12 | 9, 10, 11 | 3imtr4i 200 | 1 ⊢ ((𝜑 ⊻ 𝜓) → DECID 𝜑) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∨ wo 698 DECID wdc 824 ⊻ wxo 1364 |
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-dc 825 df-xor 1365 |
This theorem is referenced by: (None) |
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