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Theorem 2euexv 2712
Description: Double quantification with existential uniqueness. Version of 2euex 2722 with 𝑥 and 𝑦 distinct, but not requiring ax-13 2386. (Contributed by NM, 3-Dec-2001.) (Revised by Wolf Lammen, 2-Oct-2023.)
Assertion
Ref Expression
2euexv (∃!𝑥𝑦𝜑 → ∃𝑦∃!𝑥𝜑)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem 2euexv
StepHypRef Expression
1 df-eu 2650 . 2 (∃!𝑥𝑦𝜑 ↔ (∃𝑥𝑦𝜑 ∧ ∃*𝑥𝑦𝜑))
2 excom 2165 . . . 4 (∃𝑥𝑦𝜑 ↔ ∃𝑦𝑥𝜑)
3 nfe1 2150 . . . . . 6 𝑦𝑦𝜑
43nfmov 2640 . . . . 5 𝑦∃*𝑥𝑦𝜑
5 19.8a 2176 . . . . . . 7 (𝜑 → ∃𝑦𝜑)
65moimi 2623 . . . . . 6 (∃*𝑥𝑦𝜑 → ∃*𝑥𝜑)
7 moeu 2664 . . . . . 6 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
86, 7sylib 220 . . . . 5 (∃*𝑥𝑦𝜑 → (∃𝑥𝜑 → ∃!𝑥𝜑))
94, 8eximd 2212 . . . 4 (∃*𝑥𝑦𝜑 → (∃𝑦𝑥𝜑 → ∃𝑦∃!𝑥𝜑))
102, 9syl5bi 244 . . 3 (∃*𝑥𝑦𝜑 → (∃𝑥𝑦𝜑 → ∃𝑦∃!𝑥𝜑))
1110impcom 410 . 2 ((∃𝑥𝑦𝜑 ∧ ∃*𝑥𝑦𝜑) → ∃𝑦∃!𝑥𝜑)
121, 11sylbi 219 1 (∃!𝑥𝑦𝜑 → ∃𝑦∃!𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  wex 1776  ∃*wmo 2616  ∃!weu 2649
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-10 2141  ax-11 2157  ax-12 2173
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-mo 2618  df-eu 2650
This theorem is referenced by:  2exeuv  2713
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