Theorem reupick 1578

Description: Restricted uniqueness "picks" a member of a subclass.

Assertion

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

Distinct variable group(s):   x,A   x,B

Proof of Theorem reupick

Step	Hyp	Ref	Expression
1		ssel 1502	. . 3 ⊢ (A ⊆ B → (x ∈ A → x ∈ B))
2	1	ad2antll 320	. 2 ⊢ (((A ⊆ B ∧ (∃x ∈ A φ ∧ ∃!x ∈ B φ)) ∧ φ) → (x ∈ A → x ∈ B))
3	1	ancrd 247	. . . . . . . . . . . 12 ⊢ (A ⊆ B → (x ∈ A → (x ∈ B ∧ x ∈ A)))
4	3	anim1d 432	. . . . . . . . . . 11 ⊢ (A ⊆ B → ((x ∈ A ∧ φ) → ((x ∈ B ∧ x ∈ A) ∧ φ)))
5		an23 371	. . . . . . . . . . 11 ⊢ (((x ∈ B ∧ x ∈ A) ∧ φ) ↔ ((x ∈ B ∧ φ) ∧ x ∈ A))
6	4, 5	syl6ib 185	. . . . . . . . . 10 ⊢ (A ⊆ B → ((x ∈ A ∧ φ) → ((x ∈ B ∧ φ) ∧ x ∈ A)))
7	6	19.22dv 947	. . . . . . . . 9 ⊢ (A ⊆ B → (∃x(x ∈ A ∧ φ) → ∃x((x ∈ B ∧ φ) ∧ x ∈ A)))
8		eupick 1055	. . . . . . . . . 10 ⊢ ((∃!x(x ∈ B ∧ φ) ∧ ∃x((x ∈ B ∧ φ) ∧ x ∈ A)) → ((x ∈ B ∧ φ) → x ∈ A))
9	8	exp 291	. . . . . . . . 9 ⊢ (∃!x(x ∈ B ∧ φ) → (∃x((x ∈ B ∧ φ) ∧ x ∈ A) → ((x ∈ B ∧ φ) → x ∈ A)))
10	7, 9	syl9 55	. . . . . . . 8 ⊢ (A ⊆ B → (∃!x(x ∈ B ∧ φ) → (∃x(x ∈ A ∧ φ) → ((x ∈ B ∧ φ) → x ∈ A))))
11	10	com23 32	. . . . . . 7 ⊢ (A ⊆ B → (∃x(x ∈ A ∧ φ) → (∃!x(x ∈ B ∧ φ) → ((x ∈ B ∧ φ) → x ∈ A))))
12	11	imp32 281	. . . . . 6 ⊢ ((A ⊆ B ∧ (∃x(x ∈ A ∧ φ) ∧ ∃!x(x ∈ B ∧ φ))) → ((x ∈ B ∧ φ) → x ∈ A))
13		df-rex 1206	. . . . . . 7 ⊢ (∃x ∈ A φ ↔ ∃x(x ∈ A ∧ φ))
14		df-reu 1207	. . . . . . 7 ⊢ (∃!x ∈ B φ ↔ ∃!x(x ∈ B ∧ φ))
15	13, 14	anbi12i 369	. . . . . 6 ⊢ ((∃x ∈ A φ ∧ ∃!x ∈ B φ) ↔ (∃x(x ∈ A ∧ φ) ∧ ∃!x(x ∈ B ∧ φ)))
16	12, 15	sylan2b 347	. . . . 5 ⊢ ((A ⊆ B ∧ (∃x ∈ A φ ∧ ∃!x ∈ B φ)) → ((x ∈ B ∧ φ) → x ∈ A))
17	16	exp3a 292	. . . 4 ⊢ ((A ⊆ B ∧ (∃x ∈ A φ ∧ ∃!x ∈ B φ)) → (x ∈ B → (φ → x ∈ A)))
18	17	com23 32	. . 3 ⊢ ((A ⊆ B ∧ (∃x ∈ A φ ∧ ∃!x ∈ B φ)) → (φ → (x ∈ B → x ∈ A)))
19	18	imp 277	. 2 ⊢ (((A ⊆ B ∧ (∃x ∈ A φ ∧ ∃!x ∈ B φ)) ∧ φ) → (x ∈ B → x ∈ A))
20	2, 19	impbid 397	1 ⊢ (((A ⊆ B ∧ (∃x ∈ A φ ∧ ∃!x ∈ B φ)) ∧ φ) → (x ∈ A ↔ x ∈ B))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∃wex 678  ∃!weu 1007   ∈ wcel 1092  ∃wrex 1202  ∃!wreu 1203   ⊆ wss 1487

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-reu 1207  df-in 1491  df-ss 1492

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