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Theorem moexexvw 2625
Description: "At most one" double quantification. Version of moexexv 2636 with an additional disjoint variable condition, which does not require ax-13 2374. (Contributed by NM, 26-Jan-1997.) (Revised by GG, 22-Aug-2023.) Factor out common proof lines with moexex 2635. (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 1911 . 2 𝑦𝜑
2 nfv 1911 . 2 𝑦∃*𝑥𝜑
3 nfe1 2147 . . 3 𝑥𝑥(𝜑𝜓)
43nfmov 2557 . 2 𝑥∃*𝑦𝑥(𝜑𝜓)
51, 2, 4moexexlem 2623 1 ((∃*𝑥𝜑 ∧ ∀𝑥∃*𝑦𝜓) → ∃*𝑦𝑥(𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1534  wex 1775  ∃*wmo 2535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-10 2138  ax-11 2154  ax-12 2174
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1539  df-ex 1776  df-nf 1780  df-mo 2537
This theorem is referenced by:  mosub  3721  funco  6607  tfsconcatlem  43325
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