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Theorem moexexvw 2693
Description: "At most one" double quantification. Version of moexexv 2704 with an additional disjoint variable condition, which does not require ax-13 2382. (Contributed by NM, 26-Jan-1997.) (Revised by Gino Giotto, 22-Aug-2023.) Factor out common proof lines with moexex 2703. (Revised by Wolf Lammen, 2-Oct-2023.)
Assertion
Ref Expression
moexexvw ((∃*𝑥𝜑 ∧ ∀𝑥∃*𝑦𝜓) → ∃*𝑦𝑥(𝜑𝜓))
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥,𝑦)

Proof of Theorem moexexvw
StepHypRef Expression
1 nfv 1915 . 2 𝑦𝜑
2 nfv 1915 . 2 𝑦∃*𝑥𝜑
3 nfe1 2152 . . 3 𝑥𝑥(𝜑𝜓)
43nfmov 2622 . 2 𝑥∃*𝑦𝑥(𝜑𝜓)
51, 2, 4moexexlem 2691 1 ((∃*𝑥𝜑 ∧ ∀𝑥∃*𝑦𝜓) → ∃*𝑦𝑥(𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wal 1536  wex 1781  ∃*wmo 2599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2143  ax-11 2159  ax-12 2176
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601
This theorem is referenced by:  mosub  3655  funco  6368
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