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Theorem 2euswap 2536
Description: A condition allowing swap of uniqueness and existential quantifiers. (Contributed by NM, 10-Apr-2004.)
Assertion
Ref Expression
2euswap (∀𝑥∃*𝑦𝜑 → (∃!𝑥𝑦𝜑 → ∃!𝑦𝑥𝜑))

Proof of Theorem 2euswap
StepHypRef Expression
1 excomim 2030 . . . 4 (∃𝑥𝑦𝜑 → ∃𝑦𝑥𝜑)
21a1i 11 . . 3 (∀𝑥∃*𝑦𝜑 → (∃𝑥𝑦𝜑 → ∃𝑦𝑥𝜑))
3 2moswap 2535 . . 3 (∀𝑥∃*𝑦𝜑 → (∃*𝑥𝑦𝜑 → ∃*𝑦𝑥𝜑))
42, 3anim12d 584 . 2 (∀𝑥∃*𝑦𝜑 → ((∃𝑥𝑦𝜑 ∧ ∃*𝑥𝑦𝜑) → (∃𝑦𝑥𝜑 ∧ ∃*𝑦𝑥𝜑)))
5 eu5 2484 . 2 (∃!𝑥𝑦𝜑 ↔ (∃𝑥𝑦𝜑 ∧ ∃*𝑥𝑦𝜑))
6 eu5 2484 . 2 (∃!𝑦𝑥𝜑 ↔ (∃𝑦𝑥𝜑 ∧ ∃*𝑦𝑥𝜑))
74, 5, 63imtr4g 284 1 (∀𝑥∃*𝑦𝜑 → (∃!𝑥𝑦𝜑 → ∃!𝑦𝑥𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wal 1473  wex 1695  ∃!weu 2458  ∃*wmo 2459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-eu 2462  df-mo 2463
This theorem is referenced by:  2eu1  2541  euxfr2  3358  2reuswap  3377  2reuswap2  28712
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