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Mirrors > Home > ILE Home > Th. List > 2euswapdc | GIF version |
Description: A condition allowing swap of uniqueness and existential quantifiers. (Contributed by Jim Kingdon, 7-Jul-2018.) |
Ref | Expression |
---|---|
2euswapdc | ⊢ (DECID ∃𝑥∃𝑦𝜑 → (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃𝑦𝜑 → ∃!𝑦∃𝑥𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | excomim 1642 | . . . . 5 ⊢ (∃𝑥∃𝑦𝜑 → ∃𝑦∃𝑥𝜑) | |
2 | 1 | a1i 9 | . . . 4 ⊢ ((DECID ∃𝑥∃𝑦𝜑 ∧ ∀𝑥∃*𝑦𝜑) → (∃𝑥∃𝑦𝜑 → ∃𝑦∃𝑥𝜑)) |
3 | 2moswapdc 2090 | . . . . 5 ⊢ (DECID ∃𝑥∃𝑦𝜑 → (∀𝑥∃*𝑦𝜑 → (∃*𝑥∃𝑦𝜑 → ∃*𝑦∃𝑥𝜑))) | |
4 | 3 | imp 123 | . . . 4 ⊢ ((DECID ∃𝑥∃𝑦𝜑 ∧ ∀𝑥∃*𝑦𝜑) → (∃*𝑥∃𝑦𝜑 → ∃*𝑦∃𝑥𝜑)) |
5 | 2, 4 | anim12d 333 | . . 3 ⊢ ((DECID ∃𝑥∃𝑦𝜑 ∧ ∀𝑥∃*𝑦𝜑) → ((∃𝑥∃𝑦𝜑 ∧ ∃*𝑥∃𝑦𝜑) → (∃𝑦∃𝑥𝜑 ∧ ∃*𝑦∃𝑥𝜑))) |
6 | eu5 2047 | . . 3 ⊢ (∃!𝑥∃𝑦𝜑 ↔ (∃𝑥∃𝑦𝜑 ∧ ∃*𝑥∃𝑦𝜑)) | |
7 | eu5 2047 | . . 3 ⊢ (∃!𝑦∃𝑥𝜑 ↔ (∃𝑦∃𝑥𝜑 ∧ ∃*𝑦∃𝑥𝜑)) | |
8 | 5, 6, 7 | 3imtr4g 204 | . 2 ⊢ ((DECID ∃𝑥∃𝑦𝜑 ∧ ∀𝑥∃*𝑦𝜑) → (∃!𝑥∃𝑦𝜑 → ∃!𝑦∃𝑥𝜑)) |
9 | 8 | ex 114 | 1 ⊢ (DECID ∃𝑥∃𝑦𝜑 → (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃𝑦𝜑 → ∃!𝑦∃𝑥𝜑))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 DECID wdc 820 ∀wal 1330 ∃wex 1469 ∃!weu 2000 ∃*wmo 2001 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 |
This theorem is referenced by: euxfr2dc 2873 2reuswapdc 2892 |
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