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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 1472 | . . . 4 ⊢ Ⅎ𝑦∃𝑦𝜑 | |
2 | 1 | moexexdc 2081 | . . 3 ⊢ (DECID ∃𝑥∃𝑦𝜑 → ((∃*𝑥∃𝑦𝜑 ∧ ∀𝑥∃*𝑦𝜑) → ∃*𝑦∃𝑥(∃𝑦𝜑 ∧ 𝜑))) |
3 | 2 | expcomd 1417 | . 2 ⊢ (DECID ∃𝑥∃𝑦𝜑 → (∀𝑥∃*𝑦𝜑 → (∃*𝑥∃𝑦𝜑 → ∃*𝑦∃𝑥(∃𝑦𝜑 ∧ 𝜑)))) |
4 | 19.8a 1569 | . . . . . 6 ⊢ (𝜑 → ∃𝑦𝜑) | |
5 | 4 | pm4.71ri 389 | . . . . 5 ⊢ (𝜑 ↔ (∃𝑦𝜑 ∧ 𝜑)) |
6 | 5 | exbii 1584 | . . . 4 ⊢ (∃𝑥𝜑 ↔ ∃𝑥(∃𝑦𝜑 ∧ 𝜑)) |
7 | 6 | mobii 2034 | . . 3 ⊢ (∃*𝑦∃𝑥𝜑 ↔ ∃*𝑦∃𝑥(∃𝑦𝜑 ∧ 𝜑)) |
8 | 7 | imbi2i 225 | . 2 ⊢ ((∃*𝑥∃𝑦𝜑 → ∃*𝑦∃𝑥𝜑) ↔ (∃*𝑥∃𝑦𝜑 → ∃*𝑦∃𝑥(∃𝑦𝜑 ∧ 𝜑))) |
9 | 3, 8 | syl6ibr 161 | 1 ⊢ (DECID ∃𝑥∃𝑦𝜑 → (∀𝑥∃*𝑦𝜑 → (∃*𝑥∃𝑦𝜑 → ∃*𝑦∃𝑥𝜑))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 DECID wdc 819 ∀wal 1329 ∃wex 1468 ∃*wmo 1998 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 |
This theorem is referenced by: 2euswapdc 2088 |
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