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Theorem 2euswap 2577
 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 2083 . . . 4 (∃𝑥𝑦𝜑 → ∃𝑦𝑥𝜑)
21a1i 11 . . 3 (∀𝑥∃*𝑦𝜑 → (∃𝑥𝑦𝜑 → ∃𝑦𝑥𝜑))
3 2moswap 2576 . . 3 (∀𝑥∃*𝑦𝜑 → (∃*𝑥𝑦𝜑 → ∃*𝑦𝑥𝜑))
42, 3anim12d 585 . 2 (∀𝑥∃*𝑦𝜑 → ((∃𝑥𝑦𝜑 ∧ ∃*𝑥𝑦𝜑) → (∃𝑦𝑥𝜑 ∧ ∃*𝑦𝑥𝜑)))
5 eu5 2524 . 2 (∃!𝑥𝑦𝜑 ↔ (∃𝑥𝑦𝜑 ∧ ∃*𝑥𝑦𝜑))
6 eu5 2524 . 2 (∃!𝑦𝑥𝜑 ↔ (∃𝑦𝑥𝜑 ∧ ∃*𝑦𝑥𝜑))
74, 5, 63imtr4g 285 1 (∀𝑥∃*𝑦𝜑 → (∃!𝑥𝑦𝜑 → ∃!𝑦𝑥𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383  ∀wal 1521  ∃wex 1744  ∃!weu 2498  ∃*wmo 2499 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-eu 2502  df-mo 2503 This theorem is referenced by:  2eu1  2582  euxfr2  3424  2reuswap  3443  2reuswap2  29455
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