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Theorem 2moex 2705
Description: Double quantification with "at most one". Usage of this theorem is discouraged because it depends on ax-13 2382. Use the weaker 2moexv 2692 when possible. (Contributed by NM, 3-Dec-2001.) (New usage is discouraged.)
Assertion
Ref Expression
2moex (∃*𝑥𝑦𝜑 → ∀𝑦∃*𝑥𝜑)

Proof of Theorem 2moex
StepHypRef Expression
1 nfe1 2152 . . 3 𝑦𝑦𝜑
21nfmo 2624 . 2 𝑦∃*𝑥𝑦𝜑
3 19.8a 2179 . . 3 (𝜑 → ∃𝑦𝜑)
43moimi 2606 . 2 (∃*𝑥𝑦𝜑 → ∃*𝑥𝜑)
52, 4alrimi 2212 1 (∃*𝑥𝑦𝜑 → ∀𝑦∃*𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  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  ax-13 2382
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-mo 2601
This theorem is referenced by:  2eu2  2717  2eu5OLD  2721
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