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Mirrors > Home > ILE Home > Th. List > 2moswapdc | GIF version |
Description: A condition allowing swap of "at most one" and existential quantifiers. (Contributed by Jim Kingdon, 6-Jul-2018.) |
Ref | Expression |
---|---|
2moswapdc | ⊢ (DECID ∃𝑥∃𝑦𝜑 → (∀𝑥∃*𝑦𝜑 → (∃*𝑥∃𝑦𝜑 → ∃*𝑦∃𝑥𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfe1 1385 | . . . 4 ⊢ Ⅎ𝑦∃𝑦𝜑 | |
2 | 1 | moexexdc 1984 | . . 3 ⊢ (DECID ∃𝑥∃𝑦𝜑 → ((∃*𝑥∃𝑦𝜑 ∧ ∀𝑥∃*𝑦𝜑) → ∃*𝑦∃𝑥(∃𝑦𝜑 ∧ 𝜑))) |
3 | 2 | expcomd 1330 | . 2 ⊢ (DECID ∃𝑥∃𝑦𝜑 → (∀𝑥∃*𝑦𝜑 → (∃*𝑥∃𝑦𝜑 → ∃*𝑦∃𝑥(∃𝑦𝜑 ∧ 𝜑)))) |
4 | 19.8a 1482 | . . . . . 6 ⊢ (𝜑 → ∃𝑦𝜑) | |
5 | 4 | pm4.71ri 372 | . . . . 5 ⊢ (𝜑 ↔ (∃𝑦𝜑 ∧ 𝜑)) |
6 | 5 | exbii 1496 | . . . 4 ⊢ (∃𝑥𝜑 ↔ ∃𝑥(∃𝑦𝜑 ∧ 𝜑)) |
7 | 6 | mobii 1937 | . . 3 ⊢ (∃*𝑦∃𝑥𝜑 ↔ ∃*𝑦∃𝑥(∃𝑦𝜑 ∧ 𝜑)) |
8 | 7 | imbi2i 215 | . 2 ⊢ ((∃*𝑥∃𝑦𝜑 → ∃*𝑦∃𝑥𝜑) ↔ (∃*𝑥∃𝑦𝜑 → ∃*𝑦∃𝑥(∃𝑦𝜑 ∧ 𝜑))) |
9 | 3, 8 | syl6ibr 151 | 1 ⊢ (DECID ∃𝑥∃𝑦𝜑 → (∀𝑥∃*𝑦𝜑 → (∃*𝑥∃𝑦𝜑 → ∃*𝑦∃𝑥𝜑))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 DECID wdc 742 ∀wal 1241 ∃wex 1381 ∃*wmo 1901 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 |
This theorem is referenced by: 2euswapdc 1991 |
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