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Mirrors > Home > MPE Home > Th. List > 2moswap | Structured version Visualization version GIF version |
Description: A condition allowing to swap an existential quantifier and at at-most-one quantifier. (Contributed by NM, 10-Apr-2004.) |
Ref | Expression |
---|---|
2moswap | ⊢ (∀𝑥∃*𝑦𝜑 → (∃*𝑥∃𝑦𝜑 → ∃*𝑦∃𝑥𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfe1 2145 | . . . 4 ⊢ Ⅎ𝑦∃𝑦𝜑 | |
2 | 1 | moexex 2716 | . . 3 ⊢ ((∃*𝑥∃𝑦𝜑 ∧ ∀𝑥∃*𝑦𝜑) → ∃*𝑦∃𝑥(∃𝑦𝜑 ∧ 𝜑)) |
3 | 2 | expcom 414 | . 2 ⊢ (∀𝑥∃*𝑦𝜑 → (∃*𝑥∃𝑦𝜑 → ∃*𝑦∃𝑥(∃𝑦𝜑 ∧ 𝜑))) |
4 | 19.8a 2170 | . . . . 5 ⊢ (𝜑 → ∃𝑦𝜑) | |
5 | 4 | pm4.71ri 561 | . . . 4 ⊢ (𝜑 ↔ (∃𝑦𝜑 ∧ 𝜑)) |
6 | 5 | exbii 1839 | . . 3 ⊢ (∃𝑥𝜑 ↔ ∃𝑥(∃𝑦𝜑 ∧ 𝜑)) |
7 | 6 | mobii 2624 | . 2 ⊢ (∃*𝑦∃𝑥𝜑 ↔ ∃*𝑦∃𝑥(∃𝑦𝜑 ∧ 𝜑)) |
8 | 3, 7 | syl6ibr 253 | 1 ⊢ (∀𝑥∃*𝑦𝜑 → (∃*𝑥∃𝑦𝜑 → ∃*𝑦∃𝑥𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∀wal 1526 ∃wex 1771 ∃*wmo 2613 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-10 2136 ax-11 2151 ax-12 2167 ax-13 2381 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 |
This theorem is referenced by: 2euswap 2723 |
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