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Theorem 2moswap 2546
Description: A condition allowing swap of "at most one" and existential quantifiers. (Contributed by NM, 10-Apr-2004.)
Assertion
Ref Expression
2moswap (∀𝑥∃*𝑦𝜑 → (∃*𝑥𝑦𝜑 → ∃*𝑦𝑥𝜑))

Proof of Theorem 2moswap
StepHypRef Expression
1 nfe1 2026 . . . 4 𝑦𝑦𝜑
21moexex 2540 . . 3 ((∃*𝑥𝑦𝜑 ∧ ∀𝑥∃*𝑦𝜑) → ∃*𝑦𝑥(∃𝑦𝜑𝜑))
32expcom 451 . 2 (∀𝑥∃*𝑦𝜑 → (∃*𝑥𝑦𝜑 → ∃*𝑦𝑥(∃𝑦𝜑𝜑)))
4 19.8a 2051 . . . . 5 (𝜑 → ∃𝑦𝜑)
54pm4.71ri 665 . . . 4 (𝜑 ↔ (∃𝑦𝜑𝜑))
65exbii 1773 . . 3 (∃𝑥𝜑 ↔ ∃𝑥(∃𝑦𝜑𝜑))
76mobii 2492 . 2 (∃*𝑦𝑥𝜑 ↔ ∃*𝑦𝑥(∃𝑦𝜑𝜑))
83, 7syl6ibr 242 1 (∀𝑥∃*𝑦𝜑 → (∃*𝑥𝑦𝜑 → ∃*𝑦𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1480  wex 1703  ∃*wmo 2470
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1485  df-ex 1704  df-nf 1709  df-eu 2473  df-mo 2474
This theorem is referenced by:  2euswap  2547  2rmoswap  40953
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