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Theorem 2eu2 2541
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 2486 . . 3 (∃!𝑦𝑥𝜑 → ∃*𝑦𝑥𝜑)
2 2moex 2530 . . 3 (∃*𝑦𝑥𝜑 → ∀𝑥∃*𝑦𝜑)
3 2eu1 2540 . . . 4 (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))
4 simpl 471 . . . 4 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) → ∃!𝑥𝑦𝜑)
53, 4syl6bi 241 . . 3 (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 → ∃!𝑥𝑦𝜑))
61, 2, 53syl 18 . 2 (∃!𝑦𝑥𝜑 → (∃!𝑥∃!𝑦𝜑 → ∃!𝑥𝑦𝜑))
7 2exeu 2536 . . 3 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) → ∃!𝑥∃!𝑦𝜑)
87expcom 449 . 2 (∃!𝑦𝑥𝜑 → (∃!𝑥𝑦𝜑 → ∃!𝑥∃!𝑦𝜑))
96, 8impbid 200 1 (∃!𝑦𝑥𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ ∃!𝑥𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  wal 1472  wex 1694  ∃!weu 2457  ∃*wmo 2458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-eu 2461  df-mo 2462
This theorem is referenced by:  2eu8  2547
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