Theorem eu6 2604

Description: Alternate definition of the unique existential quantifier df-eu 2599 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 2599 was then proved as dfeu 2625. (Revised by BJ, 30-Sep-2022.) (Proof shortened by Wolf Lammen, 3-Jan-2023.) Remove use of ax-11 2195. (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 2565	. . . 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 2600	. . 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 2598

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 2179  ax-12 2216

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-nf 1817  df-mo 2569  df-eu 2599

This theorem is used by:  euf  2606  nfeu1ALT  2618  dfmo2  2626  sb8eulem  2628  reu6  3691  euabsn2  4693  eunex  5363  euotd  5498  iotauni  6517  iota1  6519  iotanul  6520  iota4  6521  fv3  6903  eufnfv  7234  seqomlem2  8444  aceq1  10117  dfac5  10128  bnj89  35177  cbveud  38077  wl-eudf  38286  wl-euequf  38288  wl-sb8eut  38292  wl-sb8eutv  38293  iotain  45187  iotaexeu  45188  iotasbc  45189  iotavalsb  45203  sbiota1  45204  dfac5prim  45759  permac8prim  45783  eusnsn  47823  mo0sn  49653