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Theorem 2moswap 2674
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 2144 . . . 4 𝑦𝑦𝜑
21moexex 2668 . . 3 ((∃*𝑥𝑦𝜑 ∧ ∀𝑥∃*𝑦𝜑) → ∃*𝑦𝑥(∃𝑦𝜑𝜑))
32expcom 404 . 2 (∀𝑥∃*𝑦𝜑 → (∃*𝑥𝑦𝜑 → ∃*𝑦𝑥(∃𝑦𝜑𝜑)))
4 19.8a 2166 . . . . 5 (𝜑 → ∃𝑦𝜑)
54pm4.71ri 556 . . . 4 (𝜑 ↔ (∃𝑦𝜑𝜑))
65exbii 1892 . . 3 (∃𝑥𝜑 ↔ ∃𝑥(∃𝑦𝜑𝜑))
76mobii 2563 . 2 (∃*𝑦𝑥𝜑 ↔ ∃*𝑦𝑥(∃𝑦𝜑𝜑))
83, 7syl6ibr 244 1 (∀𝑥∃*𝑦𝜑 → (∃*𝑥𝑦𝜑 → ∃*𝑦𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  wal 1599  wex 1823  ∃*wmo 2549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-mo 2551
This theorem is referenced by:  2euswap  2675  2rmoswap  42149
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