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Theorem 2eu2 2692
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 2636 . . 3 (∃!𝑦𝑥𝜑 → ∃*𝑦𝑥𝜑)
2 2moex 2681 . . 3 (∃*𝑦𝑥𝜑 → ∀𝑥∃*𝑦𝜑)
3 2eu1 2691 . . . 4 (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))
4 simpl 474 . . . 4 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) → ∃!𝑥𝑦𝜑)
53, 4syl6bi 243 . . 3 (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 → ∃!𝑥𝑦𝜑))
61, 2, 53syl 18 . 2 (∃!𝑦𝑥𝜑 → (∃!𝑥∃!𝑦𝜑 → ∃!𝑥𝑦𝜑))
7 2exeu 2687 . . 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 1630  wex 1853  ∃!weu 2607  ∃*wmo 2608
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-eu 2611  df-mo 2612
This theorem is referenced by:  2eu8  2698
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