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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 1829 | . . 3 ⊢ (∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥𝜑) | |
2 | pm2.53 848 | . . . 4 ⊢ ((𝜑 ∨ 𝜓) → (¬ 𝜑 → 𝜓)) | |
3 | 2 | aleximi 1834 | . . 3 ⊢ (∀𝑥(𝜑 ∨ 𝜓) → (∃𝑥 ¬ 𝜑 → ∃𝑥𝜓)) |
4 | 1, 3 | syl5bir 242 | . 2 ⊢ (∀𝑥(𝜑 ∨ 𝜓) → (¬ ∀𝑥𝜑 → ∃𝑥𝜓)) |
5 | 4 | orrd 860 | 1 ⊢ (∀𝑥(𝜑 ∨ 𝜓) → (∀𝑥𝜑 ∨ ∃𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 844 ∀wal 1537 ∃wex 1782 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 |
This theorem depends on definitions: df-bi 206 df-or 845 df-ex 1783 |
This theorem is referenced by: 19.33b 1888 |
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