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Theorem 2euex 2708
Description: Double quantification with existential uniqueness. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
2euex (∃!𝑥𝑦𝜑 → ∃𝑦∃!𝑥𝜑)

Proof of Theorem 2euex
StepHypRef Expression
1 eu5 2658 . 2 (∃!𝑥𝑦𝜑 ↔ (∃𝑥𝑦𝜑 ∧ ∃*𝑥𝑦𝜑))
2 excom 2211 . . . 4 (∃𝑥𝑦𝜑 ↔ ∃𝑦𝑥𝜑)
3 nfe1 2196 . . . . . 6 𝑦𝑦𝜑
43nfmo 2649 . . . . 5 𝑦∃*𝑥𝑦𝜑
5 19.8a 2219 . . . . . . 7 (𝜑 → ∃𝑦𝜑)
65moimi 2683 . . . . . 6 (∃*𝑥𝑦𝜑 → ∃*𝑥𝜑)
7 df-mo 2637 . . . . . 6 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
86, 7sylib 209 . . . . 5 (∃*𝑥𝑦𝜑 → (∃𝑥𝜑 → ∃!𝑥𝜑))
94, 8eximd 2254 . . . 4 (∃*𝑥𝑦𝜑 → (∃𝑦𝑥𝜑 → ∃𝑦∃!𝑥𝜑))
102, 9syl5bi 233 . . 3 (∃*𝑥𝑦𝜑 → (∃𝑥𝑦𝜑 → ∃𝑦∃!𝑥𝜑))
1110impcom 396 . 2 ((∃𝑥𝑦𝜑 ∧ ∃*𝑥𝑦𝜑) → ∃𝑦∃!𝑥𝜑)
121, 11sylbi 208 1 (∃!𝑥𝑦𝜑 → ∃𝑦∃!𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wex 1859  ∃!weu 2632  ∃*wmo 2633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1880  ax-4 1897  ax-5 2004  ax-6 2070  ax-7 2106  ax-10 2187  ax-11 2203  ax-12 2216  ax-13 2422
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1865  df-eu 2636  df-mo 2637
This theorem is referenced by:  2exeu  2713
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