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Theorem 2eu2 2542
 Description: Double existential uniqueness. (Contributed by NM, 3-Dec-2001.)
Assertion
Ref Expression
2eu2 (∃!𝑦𝑥𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ ∃!𝑥𝑦𝜑))

Proof of Theorem 2eu2
StepHypRef Expression
1 eumo 2487 . . 3 (∃!𝑦𝑥𝜑 → ∃*𝑦𝑥𝜑)
2 2moex 2531 . . 3 (∃*𝑦𝑥𝜑 → ∀𝑥∃*𝑦𝜑)
3 2eu1 2541 . . . 4 (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))
4 simpl 472 . . . 4 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) → ∃!𝑥𝑦𝜑)
53, 4syl6bi 242 . . 3 (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 → ∃!𝑥𝑦𝜑))
61, 2, 53syl 18 . 2 (∃!𝑦𝑥𝜑 → (∃!𝑥∃!𝑦𝜑 → ∃!𝑥𝑦𝜑))
7 2exeu 2537 . . 3 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) → ∃!𝑥∃!𝑦𝜑)
87expcom 450 . 2 (∃!𝑦𝑥𝜑 → (∃!𝑥𝑦𝜑 → ∃!𝑥∃!𝑦𝜑))
96, 8impbid 201 1 (∃!𝑦𝑥𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ ∃!𝑥𝑦𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ 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:  2eu8  2548
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