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Theorem 2euswap 2648
Description: A condition allowing to swap an existential quantifier and a unique existential quantifier. Usage of this theorem is discouraged because it depends on ax-13 2371. Use the weaker 2euswapv 2633 when possible. (Contributed by NM, 10-Apr-2004.) (New usage is discouraged.)
Assertion
Ref Expression
2euswap (∀𝑥∃*𝑦𝜑 → (∃!𝑥𝑦𝜑 → ∃!𝑦𝑥𝜑))

Proof of Theorem 2euswap
StepHypRef Expression
1 excomim 2170 . . . 4 (∃𝑥𝑦𝜑 → ∃𝑦𝑥𝜑)
21a1i 11 . . 3 (∀𝑥∃*𝑦𝜑 → (∃𝑥𝑦𝜑 → ∃𝑦𝑥𝜑))
3 2moswap 2647 . . 3 (∀𝑥∃*𝑦𝜑 → (∃*𝑥𝑦𝜑 → ∃*𝑦𝑥𝜑))
42, 3anim12d 612 . 2 (∀𝑥∃*𝑦𝜑 → ((∃𝑥𝑦𝜑 ∧ ∃*𝑥𝑦𝜑) → (∃𝑦𝑥𝜑 ∧ ∃*𝑦𝑥𝜑)))
5 df-eu 2570 . 2 (∃!𝑥𝑦𝜑 ↔ (∃𝑥𝑦𝜑 ∧ ∃*𝑥𝑦𝜑))
6 df-eu 2570 . 2 (∃!𝑦𝑥𝜑 ↔ (∃𝑦𝑥𝜑 ∧ ∃*𝑦𝑥𝜑))
74, 5, 63imtr4g 299 1 (∀𝑥∃*𝑦𝜑 → (∃!𝑥𝑦𝜑 → ∃!𝑦𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wal 1540  wex 1786  ∃*wmo 2538  ∃!weu 2569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-10 2144  ax-11 2161  ax-12 2178  ax-13 2371
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-ex 1787  df-nf 1791  df-mo 2540  df-eu 2570
This theorem is referenced by:  2eu1  2653  euxfr2  3619
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