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Theorem 2moexv 2630
Description: Double quantification with "at most one". (Contributed by NM, 3-Dec-2001.)
Assertion
Ref Expression
2moexv (∃*𝑥𝑦𝜑 → ∀𝑦∃*𝑥𝜑)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem 2moexv
StepHypRef Expression
1 nfe1 2155 . . 3 𝑦𝑦𝜑
21nfmov 2560 . 2 𝑦∃*𝑥𝑦𝜑
3 19.8a 2182 . . 3 (𝜑 → ∃𝑦𝜑)
43moimi 2545 . 2 (∃*𝑥𝑦𝜑 → ∃*𝑥𝜑)
52, 4alrimi 2215 1 (∃*𝑥𝑦𝜑 → ∀𝑦∃*𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1540  wex 1786  ∃*wmo 2538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-10 2145  ax-11 2162  ax-12 2179
This theorem depends on definitions:  df-bi 210  df-or 847  df-tru 1545  df-ex 1787  df-nf 1791  df-mo 2540
This theorem is referenced by:  2eu5  2658
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