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Mirrors > Home > MPE Home > Th. List > 2moswap | Structured version Visualization version GIF version |
Description: A condition allowing swap of "at most one" and existential quantifiers. (Contributed by NM, 10-Apr-2004.) |
Ref | Expression |
---|---|
2moswap | ⊢ (∀𝑥∃*𝑦𝜑 → (∃*𝑥∃𝑦𝜑 → ∃*𝑦∃𝑥𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfe1 2182 | . . . 4 ⊢ Ⅎ𝑦∃𝑦𝜑 | |
2 | 1 | moexex 2689 | . . 3 ⊢ ((∃*𝑥∃𝑦𝜑 ∧ ∀𝑥∃*𝑦𝜑) → ∃*𝑦∃𝑥(∃𝑦𝜑 ∧ 𝜑)) |
3 | 2 | expcom 398 | . 2 ⊢ (∀𝑥∃*𝑦𝜑 → (∃*𝑥∃𝑦𝜑 → ∃*𝑦∃𝑥(∃𝑦𝜑 ∧ 𝜑))) |
4 | 19.8a 2205 | . . . . 5 ⊢ (𝜑 → ∃𝑦𝜑) | |
5 | 4 | pm4.71ri 542 | . . . 4 ⊢ (𝜑 ↔ (∃𝑦𝜑 ∧ 𝜑)) |
6 | 5 | exbii 1923 | . . 3 ⊢ (∃𝑥𝜑 ↔ ∃𝑥(∃𝑦𝜑 ∧ 𝜑)) |
7 | 6 | mobii 2640 | . 2 ⊢ (∃*𝑦∃𝑥𝜑 ↔ ∃*𝑦∃𝑥(∃𝑦𝜑 ∧ 𝜑)) |
8 | 3, 7 | syl6ibr 242 | 1 ⊢ (∀𝑥∃*𝑦𝜑 → (∃*𝑥∃𝑦𝜑 → ∃*𝑦∃𝑥𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 ∀wal 1628 ∃wex 1851 ∃*wmo 2618 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 ax-7 2092 ax-10 2173 ax-11 2189 ax-12 2202 ax-13 2407 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-tru 1633 df-ex 1852 df-nf 1857 df-eu 2621 df-mo 2622 |
This theorem is referenced by: 2euswap 2696 2rmoswap 41698 |
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