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Theorem eu6 2571
Description: Alternate definition of the unique existential quantifier df-eu 2566 not using the at-most-one quantifier. (Contributed by NM, 12-Aug-1993.) This used to be the definition of the unique existential quantifier, while df-eu 2566 was then proved as dfeu 2592. (Revised by BJ, 30-Sep-2022.) (Proof shortened by Wolf Lammen, 3-Jan-2023.) Remove use of ax-11 2154. (Revised by SN, 21-Sep-2023.)
Assertion
Ref Expression
eu6 (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem eu6
StepHypRef Expression
1 dfmoeu 2533 . . . 4 ((∃𝑥𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦)) ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
21anbi2i 623 . . 3 ((∃𝑥𝜑 ∧ (∃𝑥𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦))) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
3 abai 827 . . 3 ((∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)) ↔ (∃𝑥𝜑 ∧ (∃𝑥𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦))))
4 eu3v 2567 . . 3 (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
52, 3, 43bitr4ri 304 . 2 (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
6 abai 827 . . 3 ((∃𝑦𝑥(𝜑𝑥 = 𝑦) ∧ ∃𝑥𝜑) ↔ (∃𝑦𝑥(𝜑𝑥 = 𝑦) ∧ (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ∃𝑥𝜑)))
7 ancom 460 . . 3 ((∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)) ↔ (∃𝑦𝑥(𝜑𝑥 = 𝑦) ∧ ∃𝑥𝜑))
8 biimpr 220 . . . . . . 7 ((𝜑𝑥 = 𝑦) → (𝑥 = 𝑦𝜑))
98alimi 1807 . . . . . 6 (∀𝑥(𝜑𝑥 = 𝑦) → ∀𝑥(𝑥 = 𝑦𝜑))
109eximi 1831 . . . . 5 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ∃𝑦𝑥(𝑥 = 𝑦𝜑))
11 exsbim 1998 . . . . 5 (∃𝑦𝑥(𝑥 = 𝑦𝜑) → ∃𝑥𝜑)
1210, 11syl 17 . . . 4 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ∃𝑥𝜑)
1312biantru 529 . . 3 (∃𝑦𝑥(𝜑𝑥 = 𝑦) ↔ (∃𝑦𝑥(𝜑𝑥 = 𝑦) ∧ (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ∃𝑥𝜑)))
146, 7, 133bitr4i 303 . 2 ((∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)) ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
155, 14bitri 275 1 (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1534  wex 1775  ∃!weu 2565
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-10 2138  ax-12 2174
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1776  df-nf 1780  df-mo 2537  df-eu 2566
This theorem is referenced by:  euf  2573  nfeu1  2585  dfmo  2593  sb8eulem  2595  reu6  3734  euabsn2  4729  eunex  5395  euotd  5522  iotauni  6537  iota1  6539  iotanul  6540  iotaexOLD  6542  iota4  6543  fv3  6924  eufnfv  7248  seqomlem2  8489  aceq1  10154  dfac5  10166  bnj89  34713  cbveud  37354  wl-eudf  37552  wl-euequf  37554  wl-sb8eut  37558  wl-sb8eutv  37559  iotain  44412  iotaexeu  44413  iotasbc  44414  iotavalsb  44428  sbiota1  44429  eusnsn  46975  mo0sn  48663
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