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Theorem 2moswap 2722
Description: A condition allowing to swap an existential quantifier and at at-most-one quantifier. (Contributed by NM, 10-Apr-2004.)
Assertion
Ref Expression
2moswap (∀𝑥∃*𝑦𝜑 → (∃*𝑥𝑦𝜑 → ∃*𝑦𝑥𝜑))

Proof of Theorem 2moswap
StepHypRef Expression
1 nfe1 2145 . . . 4 𝑦𝑦𝜑
21moexex 2716 . . 3 ((∃*𝑥𝑦𝜑 ∧ ∀𝑥∃*𝑦𝜑) → ∃*𝑦𝑥(∃𝑦𝜑𝜑))
32expcom 414 . 2 (∀𝑥∃*𝑦𝜑 → (∃*𝑥𝑦𝜑 → ∃*𝑦𝑥(∃𝑦𝜑𝜑)))
4 19.8a 2170 . . . . 5 (𝜑 → ∃𝑦𝜑)
54pm4.71ri 561 . . . 4 (𝜑 ↔ (∃𝑦𝜑𝜑))
65exbii 1839 . . 3 (∃𝑥𝜑 ↔ ∃𝑥(∃𝑦𝜑𝜑))
76mobii 2624 . 2 (∃*𝑦𝑥𝜑 ↔ ∃*𝑦𝑥(∃𝑦𝜑𝜑))
83, 7syl6ibr 253 1 (∀𝑥∃*𝑦𝜑 → (∃*𝑥𝑦𝜑 → ∃*𝑦𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1526  wex 1771  ∃*wmo 2613
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-10 2136  ax-11 2151  ax-12 2167  ax-13 2381
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615
This theorem is referenced by:  2euswap  2723
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