Theorem reu2 1338

Description: A way of expressing restricted uniqueness.

Assertion

Ref	Expression
reu2	⊢ (∃!x ∈ A φ ↔ (∃x ∈ A φ ∧ ∀x ∈ A ∀y ∈ A ((φ ∧ [y / x]φ) → x = y)))

Distinct variable group(s):   x,y,A   φ,y

Proof of Theorem reu2

Step	Hyp	Ref	Expression
1		ax-17 925	. . 3 ⊢ ((x ∈ A ∧ φ) → ∀y(x ∈ A ∧ φ))
2	1	eu2 1023	. 2 ⊢ (∃!x(x ∈ A ∧ φ) ↔ (∃x(x ∈ A ∧ φ) ∧ ∀x∀y(((x ∈ A ∧ φ) ∧ [y / x](x ∈ A ∧ φ)) → x = y)))
3		df-reu 1207	. 2 ⊢ (∃!x ∈ A φ ↔ ∃!x(x ∈ A ∧ φ))
4		df-rex 1206	. . 3 ⊢ (∃x ∈ A φ ↔ ∃x(x ∈ A ∧ φ))
5		df-ral 1205	. . . 4 ⊢ (∀x ∈ A ∀y ∈ A ((φ ∧ [y / x]φ) → x = y) ↔ ∀x(x ∈ A → ∀y ∈ A ((φ ∧ [y / x]φ) → x = y)))
6		19.21v 942	. . . . . 6 ⊢ (∀y(x ∈ A → (y ∈ A → ((φ ∧ [y / x]φ) → x = y))) ↔ (x ∈ A → ∀y(y ∈ A → ((φ ∧ [y / x]φ) → x = y))))
7		ax-17 925	. . . . . . . . . . . . 13 ⊢ (y ∈ A → ∀x y ∈ A)
8		hbs1 986	. . . . . . . . . . . . 13 ⊢ ([y / x]φ → ∀x[y / x]φ)
9	7, 8	hban 704	. . . . . . . . . . . 12 ⊢ ((y ∈ A ∧ [y / x]φ) → ∀x(y ∈ A ∧ [y / x]φ))
10		eleq1 1149	. . . . . . . . . . . . 13 ⊢ (x = y → (x ∈ A ↔ y ∈ A))
11		sbequ12 865	. . . . . . . . . . . . 13 ⊢ (x = y → (φ ↔ [y / x]φ))
12	10, 11	anbi12d 476	. . . . . . . . . . . 12 ⊢ (x = y → ((x ∈ A ∧ φ) ↔ (y ∈ A ∧ [y / x]φ)))
13	9, 12	sbie 904	. . . . . . . . . . 11 ⊢ ([y / x](x ∈ A ∧ φ) ↔ (y ∈ A ∧ [y / x]φ))
14	13	anbi2i 367	. . . . . . . . . 10 ⊢ (((x ∈ A ∧ φ) ∧ [y / x](x ∈ A ∧ φ)) ↔ ((x ∈ A ∧ φ) ∧ (y ∈ A ∧ [y / x]φ)))
15		an4 388	. . . . . . . . . 10 ⊢ (((x ∈ A ∧ φ) ∧ (y ∈ A ∧ [y / x]φ)) ↔ ((x ∈ A ∧ y ∈ A) ∧ (φ ∧ [y / x]φ)))
16	14, 15	bitr 151	. . . . . . . . 9 ⊢ (((x ∈ A ∧ φ) ∧ [y / x](x ∈ A ∧ φ)) ↔ ((x ∈ A ∧ y ∈ A) ∧ (φ ∧ [y / x]φ)))
17	16	imbi1i 161	. . . . . . . 8 ⊢ ((((x ∈ A ∧ φ) ∧ [y / x](x ∈ A ∧ φ)) → x = y) ↔ (((x ∈ A ∧ y ∈ A) ∧ (φ ∧ [y / x]φ)) → x = y))
18		impexp 276	. . . . . . . 8 ⊢ ((((x ∈ A ∧ y ∈ A) ∧ (φ ∧ [y / x]φ)) → x = y) ↔ ((x ∈ A ∧ y ∈ A) → ((φ ∧ [y / x]φ) → x = y)))
19		impexp 276	. . . . . . . 8 ⊢ (((x ∈ A ∧ y ∈ A) → ((φ ∧ [y / x]φ) → x = y)) ↔ (x ∈ A → (y ∈ A → ((φ ∧ [y / x]φ) → x = y))))
20	17, 18, 19	3bitr 155	. . . . . . 7 ⊢ ((((x ∈ A ∧ φ) ∧ [y / x](x ∈ A ∧ φ)) → x = y) ↔ (x ∈ A → (y ∈ A → ((φ ∧ [y / x]φ) → x = y))))
21	20	bial 695	. . . . . 6 ⊢ (∀y(((x ∈ A ∧ φ) ∧ [y / x](x ∈ A ∧ φ)) → x = y) ↔ ∀y(x ∈ A → (y ∈ A → ((φ ∧ [y / x]φ) → x = y))))
22		df-ral 1205	. . . . . . 7 ⊢ (∀y ∈ A ((φ ∧ [y / x]φ) → x = y) ↔ ∀y(y ∈ A → ((φ ∧ [y / x]φ) → x = y)))
23	22	imbi2i 160	. . . . . 6 ⊢ ((x ∈ A → ∀y ∈ A ((φ ∧ [y / x]φ) → x = y)) ↔ (x ∈ A → ∀y(y ∈ A → ((φ ∧ [y / x]φ) → x = y))))
24	6, 21, 23	3bitr4 158	. . . . 5 ⊢ (∀y(((x ∈ A ∧ φ) ∧ [y / x](x ∈ A ∧ φ)) → x = y) ↔ (x ∈ A → ∀y ∈ A ((φ ∧ [y / x]φ) → x = y)))
25	24	bial 695	. . . 4 ⊢ (∀x∀y(((x ∈ A ∧ φ) ∧ [y / x](x ∈ A ∧ φ)) → x = y) ↔ ∀x(x ∈ A → ∀y ∈ A ((φ ∧ [y / x]φ) → x = y)))
26	5, 25	bitr4 154	. . 3 ⊢ (∀x ∈ A ∀y ∈ A ((φ ∧ [y / x]φ) → x = y) ↔ ∀x∀y(((x ∈ A ∧ φ) ∧ [y / x](x ∈ A ∧ φ)) → x = y))
27	4, 26	anbi12i 369	. 2 ⊢ ((∃x ∈ A φ ∧ ∀x ∈ A ∀y ∈ A ((φ ∧ [y / x]φ) → x = y)) ↔ (∃x(x ∈ A ∧ φ) ∧ ∀x∀y(((x ∈ A ∧ φ) ∧ [y / x](x ∈ A ∧ φ)) → x = y)))
28	2, 3, 27	3bitr4 158	1 ⊢ (∃!x ∈ A φ ↔ (∃x ∈ A φ ∧ ∀x ∈ A ∀y ∈ A ((φ ∧ [y / x]φ) → x = y)))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678   = weq 797  [wsb 852  ∃!weu 1007   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202  ∃!wreu 1203

This theorem is referenced by:  reu4 1340

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-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-reu 1207

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