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Mirrors > Home > MPE Home > Th. List > Mathboxes > pm11.61 | Structured version Visualization version GIF version |
Description: Theorem *11.61 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.) |
Ref | Expression |
---|---|
pm11.61 | ⊢ (∃𝑦∀𝑥(𝜑 → 𝜓) → ∀𝑥(𝜑 → ∃𝑦𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.12 2320 | . 2 ⊢ (∃𝑦∀𝑥(𝜑 → 𝜓) → ∀𝑥∃𝑦(𝜑 → 𝜓)) | |
2 | 19.37v 1995 | . . . 4 ⊢ (∃𝑦(𝜑 → 𝜓) ↔ (𝜑 → ∃𝑦𝜓)) | |
3 | 2 | biimpi 215 | . . 3 ⊢ (∃𝑦(𝜑 → 𝜓) → (𝜑 → ∃𝑦𝜓)) |
4 | 3 | alimi 1813 | . 2 ⊢ (∀𝑥∃𝑦(𝜑 → 𝜓) → ∀𝑥(𝜑 → ∃𝑦𝜓)) |
5 | 1, 4 | syl 17 | 1 ⊢ (∃𝑦∀𝑥(𝜑 → 𝜓) → ∀𝑥(𝜑 → ∃𝑦𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1539 ∃wex 1781 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-10 2137 ax-11 2154 ax-12 2171 |
This theorem depends on definitions: df-bi 206 df-or 846 df-ex 1782 df-nf 1786 |
This theorem is referenced by: (None) |
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