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Mirrors > Home > MPE Home > Th. List > Mathboxes > pm10.57 | Structured version Visualization version GIF version |
Description: Theorem *10.57 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.) |
Ref | Expression |
---|---|
pm10.57 | ⊢ (∀𝑥(𝜑 → (𝜓 ∨ 𝜒)) → (∀𝑥(𝜑 → 𝜓) ∨ ∃𝑥(𝜑 ∧ 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alnex 1788 | . . . 4 ⊢ (∀𝑥 ¬ (𝜑 ∧ 𝜒) ↔ ¬ ∃𝑥(𝜑 ∧ 𝜒)) | |
2 | imnan 403 | . . . . . 6 ⊢ ((𝜑 → ¬ 𝜒) ↔ ¬ (𝜑 ∧ 𝜒)) | |
3 | pm2.53 850 | . . . . . . . 8 ⊢ ((𝜓 ∨ 𝜒) → (¬ 𝜓 → 𝜒)) | |
4 | 3 | con1d 147 | . . . . . . 7 ⊢ ((𝜓 ∨ 𝜒) → (¬ 𝜒 → 𝜓)) |
5 | 4 | imim3i 64 | . . . . . 6 ⊢ ((𝜑 → (𝜓 ∨ 𝜒)) → ((𝜑 → ¬ 𝜒) → (𝜑 → 𝜓))) |
6 | 2, 5 | syl5bir 246 | . . . . 5 ⊢ ((𝜑 → (𝜓 ∨ 𝜒)) → (¬ (𝜑 ∧ 𝜒) → (𝜑 → 𝜓))) |
7 | 6 | al2imi 1822 | . . . 4 ⊢ (∀𝑥(𝜑 → (𝜓 ∨ 𝜒)) → (∀𝑥 ¬ (𝜑 ∧ 𝜒) → ∀𝑥(𝜑 → 𝜓))) |
8 | 1, 7 | syl5bir 246 | . . 3 ⊢ (∀𝑥(𝜑 → (𝜓 ∨ 𝜒)) → (¬ ∃𝑥(𝜑 ∧ 𝜒) → ∀𝑥(𝜑 → 𝜓))) |
9 | 8 | con1d 147 | . 2 ⊢ (∀𝑥(𝜑 → (𝜓 ∨ 𝜒)) → (¬ ∀𝑥(𝜑 → 𝜓) → ∃𝑥(𝜑 ∧ 𝜒))) |
10 | 9 | orrd 862 | 1 ⊢ (∀𝑥(𝜑 → (𝜓 ∨ 𝜒)) → (∀𝑥(𝜑 → 𝜓) ∨ ∃𝑥(𝜑 ∧ 𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∨ wo 846 ∀wal 1540 ∃wex 1786 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-ex 1787 |
This theorem is referenced by: (None) |
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