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Theorem 2moswapdc 2033
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 1426 . . . 4 𝑦𝑦𝜑
21moexexdc 2027 . . 3 (DECID𝑥𝑦𝜑 → ((∃*𝑥𝑦𝜑 ∧ ∀𝑥∃*𝑦𝜑) → ∃*𝑦𝑥(∃𝑦𝜑𝜑)))
32expcomd 1371 . 2 (DECID𝑥𝑦𝜑 → (∀𝑥∃*𝑦𝜑 → (∃*𝑥𝑦𝜑 → ∃*𝑦𝑥(∃𝑦𝜑𝜑))))
4 19.8a 1523 . . . . . 6 (𝜑 → ∃𝑦𝜑)
54pm4.71ri 384 . . . . 5 (𝜑 ↔ (∃𝑦𝜑𝜑))
65exbii 1537 . . . 4 (∃𝑥𝜑 ↔ ∃𝑥(∃𝑦𝜑𝜑))
76mobii 1980 . . 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 776  wal 1283  wex 1422  ∃*wmo 1944
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-dc 777  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947
This theorem is referenced by:  2euswapdc  2034
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