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Theorem 2moswap 2732
Description: A condition allowing to swap an existential quantifier and at at-most-one quantifier. Usage of this theorem is discouraged because it depends on ax-13 2392. Use the weaker 2moswapv 2717 when possible. (Contributed by NM, 10-Apr-2004.) (New usage is discouraged.)
Assertion
Ref Expression
2moswap (∀𝑥∃*𝑦𝜑 → (∃*𝑥𝑦𝜑 → ∃*𝑦𝑥𝜑))

Proof of Theorem 2moswap
StepHypRef Expression
1 nfe1 2155 . . . 4 𝑦𝑦𝜑
21moexex 2726 . . 3 ((∃*𝑥𝑦𝜑 ∧ ∀𝑥∃*𝑦𝜑) → ∃*𝑦𝑥(∃𝑦𝜑𝜑))
32expcom 417 . 2 (∀𝑥∃*𝑦𝜑 → (∃*𝑥𝑦𝜑 → ∃*𝑦𝑥(∃𝑦𝜑𝜑)))
4 19.8a 2182 . . . . 5 (𝜑 → ∃𝑦𝜑)
54pm4.71ri 564 . . . 4 (𝜑 ↔ (∃𝑦𝜑𝜑))
65exbii 1849 . . 3 (∃𝑥𝜑 ↔ ∃𝑥(∃𝑦𝜑𝜑))
76mobii 2632 . 2 (∃*𝑦𝑥𝜑 ↔ ∃*𝑦𝑥(∃𝑦𝜑𝜑))
83, 7syl6ibr 255 1 (∀𝑥∃*𝑦𝜑 → (∃*𝑥𝑦𝜑 → ∃*𝑦𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wal 1536  wex 1781  ∃*wmo 2622
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 1912  ax-6 1971  ax-7 2016  ax-10 2146  ax-11 2162  ax-12 2179  ax-13 2392
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 2071  df-mo 2624
This theorem is referenced by:  2euswap  2733
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