Theorem reu2 3025

Description: A way to express restricted uniqueness. (Contributed by NM, 22-Nov-1994.)

Assertion

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

Distinct variable groups:   x,y,A   φ,y

Allowed substitution hint:   φ(x)

Proof of Theorem reu2

Step	Hyp	Ref	Expression
1		nfv 1619	. . 3 ⊢ Ⅎy(x ∈ A ∧ φ)
2	1	eu2 2229	. 2 ⊢ (∃!x(x ∈ A ∧ φ) ↔ (∃x(x ∈ A ∧ φ) ∧ ∀x∀y(((x ∈ A ∧ φ) ∧ [y / x](x ∈ A ∧ φ)) → x = y)))
3		df-reu 2622	. 2 ⊢ (∃!x ∈ A φ ↔ ∃!x(x ∈ A ∧ φ))
4		df-rex 2621	. . 3 ⊢ (∃x ∈ A φ ↔ ∃x(x ∈ A ∧ φ))
5		df-ral 2620	. . . 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 1890	. . . . . 6 ⊢ (∀y(x ∈ A → (y ∈ A → ((φ ∧ [y / x]φ) → x = y))) ↔ (x ∈ A → ∀y(y ∈ A → ((φ ∧ [y / x]φ) → x = y))))
7		nfv 1619	. . . . . . . . . . . . 13 ⊢ Ⅎx y ∈ A
8		nfs1v 2106	. . . . . . . . . . . . 13 ⊢ Ⅎx[y / x]φ
9	7, 8	nfan 1824	. . . . . . . . . . . 12 ⊢ Ⅎx(y ∈ A ∧ [y / x]φ)
10		eleq1 2413	. . . . . . . . . . . . 13 ⊢ (x = y → (x ∈ A ↔ y ∈ A))
11		sbequ12 1919	. . . . . . . . . . . . 13 ⊢ (x = y → (φ ↔ [y / x]φ))
12	10, 11	anbi12d 691	. . . . . . . . . . . 12 ⊢ (x = y → ((x ∈ A ∧ φ) ↔ (y ∈ A ∧ [y / x]φ)))
13	9, 12	sbie 2038	. . . . . . . . . . 11 ⊢ ([y / x](x ∈ A ∧ φ) ↔ (y ∈ A ∧ [y / x]φ))
14	13	anbi2i 675	. . . . . . . . . 10 ⊢ (((x ∈ A ∧ φ) ∧ [y / x](x ∈ A ∧ φ)) ↔ ((x ∈ A ∧ φ) ∧ (y ∈ A ∧ [y / x]φ)))
15		an4 797	. . . . . . . . . 10 ⊢ (((x ∈ A ∧ φ) ∧ (y ∈ A ∧ [y / x]φ)) ↔ ((x ∈ A ∧ y ∈ A) ∧ (φ ∧ [y / x]φ)))
16	14, 15	bitri 240	. . . . . . . . 9 ⊢ (((x ∈ A ∧ φ) ∧ [y / x](x ∈ A ∧ φ)) ↔ ((x ∈ A ∧ y ∈ A) ∧ (φ ∧ [y / x]φ)))
17	16	imbi1i 315	. . . . . . . 8 ⊢ ((((x ∈ A ∧ φ) ∧ [y / x](x ∈ A ∧ φ)) → x = y) ↔ (((x ∈ A ∧ y ∈ A) ∧ (φ ∧ [y / x]φ)) → x = y))
18		impexp 433	. . . . . . . 8 ⊢ ((((x ∈ A ∧ y ∈ A) ∧ (φ ∧ [y / x]φ)) → x = y) ↔ ((x ∈ A ∧ y ∈ A) → ((φ ∧ [y / x]φ) → x = y)))
19		impexp 433	. . . . . . . 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	3bitri 262	. . . . . . 7 ⊢ ((((x ∈ A ∧ φ) ∧ [y / x](x ∈ A ∧ φ)) → x = y) ↔ (x ∈ A → (y ∈ A → ((φ ∧ [y / x]φ) → x = y))))
21	20	albii 1566	. . . . . 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 2620	. . . . . . 7 ⊢ (∀y ∈ A ((φ ∧ [y / x]φ) → x = y) ↔ ∀y(y ∈ A → ((φ ∧ [y / x]φ) → x = y)))
23	22	imbi2i 303	. . . . . 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	3bitr4i 268	. . . . 5 ⊢ (∀y(((x ∈ A ∧ φ) ∧ [y / x](x ∈ A ∧ φ)) → x = y) ↔ (x ∈ A → ∀y ∈ A ((φ ∧ [y / x]φ) → x = y)))
25	24	albii 1566	. . . 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	bitr4i 243	. . 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 678	. 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	3bitr4i 268	1 ⊢ (∃!x ∈ A φ ↔ (∃x ∈ A φ ∧ ∀x ∈ A ∀y ∈ A ((φ ∧ [y / x]φ) → x = y)))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642  [wsb 1648   ∈ wcel 1710  ∃!weu 2204  ∀wral 2615  ∃wrex 2616  ∃!wreu 2617

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-cleq 2346  df-clel 2349  df-ral 2620  df-rex 2621  df-reu 2622

This theorem is referenced by: (None)