Theorem eu6 2599

Description: Alternate definition of the unique existential quantifier df-eu 2594 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 2594 was then proved as dfeu 2620. (Revised by BJ, 30-Sep-2022.) (Proof shortened by Wolf Lammen, 3-Jan-2023.) Remove use of ax-11 2194. (Revised by SN, 21-Sep-2023.)

Assertion

Ref	Expression
eu6	⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))

Distinct variable groups:   𝑥,𝑦   𝜑,𝑦

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem eu6

Step	Hyp	Ref	Expression
1		dfmoeu 2560	. . . 4 ⊢ ((∃𝑥𝜑 → ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
2	1	anbi2i 635	. . 3 ⊢ ((∃𝑥𝜑 ∧ (∃𝑥𝜑 → ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))) ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))
3		abai 839	. . 3 ⊢ ((∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) ↔ (∃𝑥𝜑 ∧ (∃𝑥𝜑 → ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))))
4		eu3v 2595	. . 3 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))
5	2, 3, 4	3bitr4ri 307	. 2 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))
6		abai 839	. . 3 ⊢ ((∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ ∃𝑥𝜑) ↔ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∃𝑥𝜑)))
7		ancom 466	. . 3 ⊢ ((∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) ↔ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ ∃𝑥𝜑))
8		biimpr 223	. . . . . . 7 ⊢ ((𝜑 ↔ 𝑥 = 𝑦) → (𝑥 = 𝑦 → 𝜑))
9	8	alimi 1844	. . . . . 6 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∀𝑥(𝑥 = 𝑦 → 𝜑))
10	9	eximi 1868	. . . . 5 ⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑))
11		exsbim 2035	. . . . 5 ⊢ (∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥𝜑)
12	10, 11	syl 18	. . . 4 ⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∃𝑥𝜑)
13	12	biantru 539	. . 3 ⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∃𝑥𝜑)))
14	6, 7, 13	3bitr4i 306	. 2 ⊢ ((∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
15	5, 14	bitri 278	1 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568  ∃wex 1812  ∃!weu 2593

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2178  ax-12 2213

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-nf 1817  df-mo 2564  df-eu 2594

This theorem is used by:  euf  2601  nfeu1ALT  2613  dfmo2  2621  sb8eulem  2623  reu6  3684  euabsn2  4686  eunex  5355  euotd  5490  iotauni  6510  iota1  6512  iotanul  6513  iota4  6514  fv3  6897  eufnfv  7229  seqomlem2  8443  aceq1  10123  dfac5  10134  bnj89  35234  cbveud  38129  wl-eudf  38338  wl-euequf  38340  wl-sb8eut  38344  wl-sb8eutv  38345  iotain  45244  iotaexeu  45245  iotasbc  45246  iotavalsb  45260  sbiota1  45261  dfac5prim  45816  permac8prim  45840  eusnsn  47917  mo0sn  49747