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Theorem 2moswapdc 2006
Description: A condition allowing swap of "at most one" and existential quantifiers. (Contributed by Jim Kingdon, 6-Jul-2018.)
Assertion
Ref Expression
2moswapdc (DECID𝑥𝑦𝜑 → (∀𝑥∃*𝑦𝜑 → (∃*𝑥𝑦𝜑 → ∃*𝑦𝑥𝜑)))

Proof of Theorem 2moswapdc
StepHypRef Expression
1 nfe1 1401 . . . 4 𝑦𝑦𝜑
21moexexdc 2000 . . 3 (DECID𝑥𝑦𝜑 → ((∃*𝑥𝑦𝜑 ∧ ∀𝑥∃*𝑦𝜑) → ∃*𝑦𝑥(∃𝑦𝜑𝜑)))
32expcomd 1346 . 2 (DECID𝑥𝑦𝜑 → (∀𝑥∃*𝑦𝜑 → (∃*𝑥𝑦𝜑 → ∃*𝑦𝑥(∃𝑦𝜑𝜑))))
4 19.8a 1498 . . . . . 6 (𝜑 → ∃𝑦𝜑)
54pm4.71ri 378 . . . . 5 (𝜑 ↔ (∃𝑦𝜑𝜑))
65exbii 1512 . . . 4 (∃𝑥𝜑 ↔ ∃𝑥(∃𝑦𝜑𝜑))
76mobii 1953 . . 3 (∃*𝑦𝑥𝜑 ↔ ∃*𝑦𝑥(∃𝑦𝜑𝜑))
87imbi2i 219 . 2 ((∃*𝑥𝑦𝜑 → ∃*𝑦𝑥𝜑) ↔ (∃*𝑥𝑦𝜑 → ∃*𝑦𝑥(∃𝑦𝜑𝜑)))
93, 8syl6ibr 155 1 (DECID𝑥𝑦𝜑 → (∀𝑥∃*𝑦𝜑 → (∃*𝑥𝑦𝜑 → ∃*𝑦𝑥𝜑)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  DECID wdc 753  wal 1257  wex 1397  ∃*wmo 1917
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444
This theorem depends on definitions:  df-bi 114  df-dc 754  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920
This theorem is referenced by:  2euswapdc  2007
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