Theorem reupick 3215

Description: Restricted uniqueness "picks" a member of a subclass. (Contributed by NM, 21-Aug-1999.)

Assertion

Ref	Expression
reupick	⊢ (((A ⊆ B ∧ (∃x ∈ A φ ∧ ∃!x ∈ B φ)) ∧ φ) → (x ∈ A ↔ x ∈ B))

Distinct variable groups:   x,A   x,B

Allowed substitution hint:   φ(x)

Proof of Theorem reupick

Step	Hyp	Ref	Expression
1		ssel 2933	. . 3 ⊢ (A ⊆ B → (x ∈ A → x ∈ B))
2	1	ad2antrr 457	. 2 ⊢ (((A ⊆ B ∧ (∃x ∈ A φ ∧ ∃!x ∈ B φ)) ∧ φ) → (x ∈ A → x ∈ B))
3		df-rex 2306	. . . . . 6 ⊢ (∃x ∈ A φ ↔ ∃x(x ∈ A ∧ φ))
4		df-reu 2307	. . . . . 6 ⊢ (∃!x ∈ B φ ↔ ∃!x(x ∈ B ∧ φ))
5	3, 4	anbi12i 433	. . . . 5 ⊢ ((∃x ∈ A φ ∧ ∃!x ∈ B φ) ↔ (∃x(x ∈ A ∧ φ) ∧ ∃!x(x ∈ B ∧ φ)))
6	1	ancrd 309	. . . . . . . . . . 11 ⊢ (A ⊆ B → (x ∈ A → (x ∈ B ∧ x ∈ A)))
7	6	anim1d 319	. . . . . . . . . 10 ⊢ (A ⊆ B → ((x ∈ A ∧ φ) → ((x ∈ B ∧ x ∈ A) ∧ φ)))
8		an32 496	. . . . . . . . . 10 ⊢ (((x ∈ B ∧ x ∈ A) ∧ φ) ↔ ((x ∈ B ∧ φ) ∧ x ∈ A))
9	7, 8	syl6ib 150	. . . . . . . . 9 ⊢ (A ⊆ B → ((x ∈ A ∧ φ) → ((x ∈ B ∧ φ) ∧ x ∈ A)))
10	9	eximdv 1757	. . . . . . . 8 ⊢ (A ⊆ B → (∃x(x ∈ A ∧ φ) → ∃x((x ∈ B ∧ φ) ∧ x ∈ A)))
11		eupick 1976	. . . . . . . . 9 ⊢ ((∃!x(x ∈ B ∧ φ) ∧ ∃x((x ∈ B ∧ φ) ∧ x ∈ A)) → ((x ∈ B ∧ φ) → x ∈ A))
12	11	ex 108	. . . . . . . 8 ⊢ (∃!x(x ∈ B ∧ φ) → (∃x((x ∈ B ∧ φ) ∧ x ∈ A) → ((x ∈ B ∧ φ) → x ∈ A)))
13	10, 12	syl9 66	. . . . . . 7 ⊢ (A ⊆ B → (∃!x(x ∈ B ∧ φ) → (∃x(x ∈ A ∧ φ) → ((x ∈ B ∧ φ) → x ∈ A))))
14	13	com23 72	. . . . . 6 ⊢ (A ⊆ B → (∃x(x ∈ A ∧ φ) → (∃!x(x ∈ B ∧ φ) → ((x ∈ B ∧ φ) → x ∈ A))))
15	14	imp32 244	. . . . 5 ⊢ ((A ⊆ B ∧ (∃x(x ∈ A ∧ φ) ∧ ∃!x(x ∈ B ∧ φ))) → ((x ∈ B ∧ φ) → x ∈ A))
16	5, 15	sylan2b 271	. . . 4 ⊢ ((A ⊆ B ∧ (∃x ∈ A φ ∧ ∃!x ∈ B φ)) → ((x ∈ B ∧ φ) → x ∈ A))
17	16	expcomd 1327	. . 3 ⊢ ((A ⊆ B ∧ (∃x ∈ A φ ∧ ∃!x ∈ B φ)) → (φ → (x ∈ B → x ∈ A)))
18	17	imp 115	. 2 ⊢ (((A ⊆ B ∧ (∃x ∈ A φ ∧ ∃!x ∈ B φ)) ∧ φ) → (x ∈ B → x ∈ A))
19	2, 18	impbid 120	1 ⊢ (((A ⊆ B ∧ (∃x ∈ A φ ∧ ∃!x ∈ B φ)) ∧ φ) → (x ∈ A ↔ x ∈ B))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∃wex 1378   ∈ wcel 1390  ∃!weu 1897  ∃wrex 2301  ∃!wreu 2302   ⊆ wss 2911

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-rex 2306  df-reu 2307  df-in 2918  df-ss 2925

This theorem is referenced by: (None)