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Theorem 2euex 2703
 Description: Double quantification with existential uniqueness. Usage of this theorem is discouraged because it depends on ax-13 2379. Use the weaker 2euexv 2693 when possible. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (New usage is discouraged.)
Assertion
Ref Expression
2euex (∃!𝑥𝑦𝜑 → ∃𝑦∃!𝑥𝜑)

Proof of Theorem 2euex
StepHypRef Expression
1 df-eu 2629 . 2 (∃!𝑥𝑦𝜑 ↔ (∃𝑥𝑦𝜑 ∧ ∃*𝑥𝑦𝜑))
2 excom 2166 . . . 4 (∃𝑥𝑦𝜑 ↔ ∃𝑦𝑥𝜑)
3 nfe1 2151 . . . . . 6 𝑦𝑦𝜑
43nfmo 2621 . . . . 5 𝑦∃*𝑥𝑦𝜑
5 19.8a 2178 . . . . . . 7 (𝜑 → ∃𝑦𝜑)
65moimi 2603 . . . . . 6 (∃*𝑥𝑦𝜑 → ∃*𝑥𝜑)
7 moeu 2643 . . . . . 6 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
86, 7sylib 221 . . . . 5 (∃*𝑥𝑦𝜑 → (∃𝑥𝜑 → ∃!𝑥𝜑))
94, 8eximd 2214 . . . 4 (∃*𝑥𝑦𝜑 → (∃𝑦𝑥𝜑 → ∃𝑦∃!𝑥𝜑))
102, 9syl5bi 245 . . 3 (∃*𝑥𝑦𝜑 → (∃𝑥𝑦𝜑 → ∃𝑦∃!𝑥𝜑))
1110impcom 411 . 2 ((∃𝑥𝑦𝜑 ∧ ∃*𝑥𝑦𝜑) → ∃𝑦∃!𝑥𝜑)
121, 11sylbi 220 1 (∃!𝑥𝑦𝜑 → ∃𝑦∃!𝑥𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399  ∃wex 1781  ∃*wmo 2596  ∃!weu 2628 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-11 2158  ax-12 2175  ax-13 2379 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-mo 2598  df-eu 2629 This theorem is referenced by:  2exeu  2708
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