Theorem eu6 2567

Description: Alternate definition of the unique existential quantifier df-eu 2562 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 2562 was then proved as dfeu 2588. (Revised by BJ, 30-Sep-2022.) (Proof shortened by Wolf Lammen, 3-Jan-2023.) Remove use of ax-11 2158. (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 2529	. . . 4 ⊢ ((∃𝑥𝜑 → ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
2	1	anbi2i 623	. . 3 ⊢ ((∃𝑥𝜑 ∧ (∃𝑥𝜑 → ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))) ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))
3		abai 826	. . 3 ⊢ ((∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) ↔ (∃𝑥𝜑 ∧ (∃𝑥𝜑 → ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))))
4		eu3v 2563	. . 3 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))
5	2, 3, 4	3bitr4ri 304	. 2 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))
6		abai 826	. . 3 ⊢ ((∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ ∃𝑥𝜑) ↔ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∃𝑥𝜑)))
7		ancom 460	. . 3 ⊢ ((∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) ↔ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ ∃𝑥𝜑))
8		biimpr 220	. . . . . . 7 ⊢ ((𝜑 ↔ 𝑥 = 𝑦) → (𝑥 = 𝑦 → 𝜑))
9	8	alimi 1811	. . . . . 6 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∀𝑥(𝑥 = 𝑦 → 𝜑))
10	9	eximi 1835	. . . . 5 ⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑))
11		exsbim 2002	. . . . 5 ⊢ (∃𝑦∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥𝜑)
12	10, 11	syl 17	. . . 4 ⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∃𝑥𝜑)
13	12	biantru 529	. . 3 ⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∃𝑥𝜑)))
14	6, 7, 13	3bitr4i 303	. 2 ⊢ ((∃𝑥𝜑 ∧ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)) ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
15	5, 14	bitri 275	1 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538  ∃wex 1779  ∃!weu 2561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-10 2142  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1780  df-nf 1784  df-mo 2533  df-eu 2562

This theorem is referenced by:  euf  2569  nfeu1  2581  dfmo  2589  sb8eulem  2591  reu6  3686  euabsn2  4677  eunex  5329  euotd  5456  iotauni  6459  iota1  6461  iotanul  6462  iota4  6463  fv3  6840  eufnfv  7165  seqomlem2  8373  aceq1  10011  dfac5  10023  bnj89  34688  cbveud  37350  wl-eudf  37550  wl-euequf  37552  wl-sb8eut  37556  wl-sb8eutv  37557  iotain  44394  iotaexeu  44395  iotasbc  44396  iotavalsb  44410  sbiota1  44411  dfac5prim  44968  permac8prim  44992  eusnsn  47014  mo0sn  48804