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Theorem 2moswapdc 2038
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 1430 . . . 4 𝑦𝑦𝜑
21moexexdc 2032 . . 3 (DECID𝑥𝑦𝜑 → ((∃*𝑥𝑦𝜑 ∧ ∀𝑥∃*𝑦𝜑) → ∃*𝑦𝑥(∃𝑦𝜑𝜑)))
32expcomd 1375 . 2 (DECID𝑥𝑦𝜑 → (∀𝑥∃*𝑦𝜑 → (∃*𝑥𝑦𝜑 → ∃*𝑦𝑥(∃𝑦𝜑𝜑))))
4 19.8a 1527 . . . . . 6 (𝜑 → ∃𝑦𝜑)
54pm4.71ri 384 . . . . 5 (𝜑 ↔ (∃𝑦𝜑𝜑))
65exbii 1541 . . . 4 (∃𝑥𝜑 ↔ ∃𝑥(∃𝑦𝜑𝜑))
76mobii 1985 . . 3 (∃*𝑦𝑥𝜑 ↔ ∃*𝑦𝑥(∃𝑦𝜑𝜑))
87imbi2i 224 . 2 ((∃*𝑥𝑦𝜑 → ∃*𝑦𝑥𝜑) ↔ (∃*𝑥𝑦𝜑 → ∃*𝑦𝑥(∃𝑦𝜑𝜑)))
93, 8syl6ibr 160 1 (DECID𝑥𝑦𝜑 → (∀𝑥∃*𝑦𝜑 → (∃*𝑥𝑦𝜑 → ∃*𝑦𝑥𝜑)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  DECID wdc 780  wal 1287  wex 1426  ∃*wmo 1949
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473
This theorem depends on definitions:  df-bi 115  df-dc 781  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952
This theorem is referenced by:  2euswapdc  2039
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