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Mirrors > Home > MPE Home > Th. List > 19.30 | Structured version Visualization version GIF version |
Description: Theorem 19.30 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Ref | Expression |
---|---|
19.30 | ⊢ (∀𝑥(𝜑 ∨ 𝜓) → (∀𝑥𝜑 ∨ ∃𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exnal 1833 | . . 3 ⊢ (∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥𝜑) | |
2 | pm2.53 850 | . . . 4 ⊢ ((𝜑 ∨ 𝜓) → (¬ 𝜑 → 𝜓)) | |
3 | 2 | aleximi 1838 | . . 3 ⊢ (∀𝑥(𝜑 ∨ 𝜓) → (∃𝑥 ¬ 𝜑 → ∃𝑥𝜓)) |
4 | 1, 3 | syl5bir 246 | . 2 ⊢ (∀𝑥(𝜑 ∨ 𝜓) → (¬ ∀𝑥𝜑 → ∃𝑥𝜓)) |
5 | 4 | orrd 862 | 1 ⊢ (∀𝑥(𝜑 ∨ 𝜓) → (∀𝑥𝜑 ∨ ∃𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ 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-or 847 df-ex 1787 |
This theorem is referenced by: 19.33b 1892 |
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