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Theorem 2eu2 2583
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 2527 . . 3 (∃!𝑦𝑥𝜑 → ∃*𝑦𝑥𝜑)
2 2moex 2572 . . 3 (∃*𝑦𝑥𝜑 → ∀𝑥∃*𝑦𝜑)
3 2eu1 2582 . . . 4 (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))
4 simpl 472 . . . 4 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) → ∃!𝑥𝑦𝜑)
53, 4syl6bi 243 . . 3 (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 → ∃!𝑥𝑦𝜑))
61, 2, 53syl 18 . 2 (∃!𝑦𝑥𝜑 → (∃!𝑥∃!𝑦𝜑 → ∃!𝑥𝑦𝜑))
7 2exeu 2578 . . 3 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) → ∃!𝑥∃!𝑦𝜑)
87expcom 450 . 2 (∃!𝑦𝑥𝜑 → (∃!𝑥𝑦𝜑 → ∃!𝑥∃!𝑦𝜑))
96, 8impbid 202 1 (∃!𝑦𝑥𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ ∃!𝑥𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  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:  2eu8  2589
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