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Theorem moexex 2722
Description: "At most one" double quantification. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 28-Dec-2018.) Factor out common proof lines with moexexvw 2712. (Revised by Wolf Lammen, 2-Oct-2023.)
Hypothesis
Ref Expression
moexex.1 𝑦𝜑
Assertion
Ref Expression
moexex ((∃*𝑥𝜑 ∧ ∀𝑥∃*𝑦𝜓) → ∃*𝑦𝑥(𝜑𝜓))

Proof of Theorem moexex
StepHypRef Expression
1 moexex.1 . 2 𝑦𝜑
21nfmo 2644 . 2 𝑦∃*𝑥𝜑
3 nfe1 2147 . . 3 𝑥𝑥(𝜑𝜓)
43nfmo 2644 . 2 𝑥∃*𝑦𝑥(𝜑𝜓)
51, 2, 4moexexlem 2710 1 ((∃*𝑥𝜑 ∧ ∀𝑥∃*𝑦𝜓) → ∃*𝑦𝑥(𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1528  wex 1773  wnf 1777  ∃*wmo 2618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620
This theorem is referenced by:  moexexv  2723  2moswap  2728
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