Theorem reu3 3027

Description: A way to express restricted uniqueness. (Contributed by NM, 24-Oct-2006.)

Assertion

Ref	Expression
reu3	⊢ (∃!x ∈ A φ ↔ (∃x ∈ A φ ∧ ∃y ∈ A ∀x ∈ A (φ → x = y)))

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

Allowed substitution hint:   φ(x)

Proof of Theorem reu3

Step	Hyp	Ref	Expression
1		reurex 2826	. . 3 ⊢ (∃!x ∈ A φ → ∃x ∈ A φ)
2		reu6 3026	. . . 4 ⊢ (∃!x ∈ A φ ↔ ∃y ∈ A ∀x ∈ A (φ ↔ x = y))
3		bi1 178	. . . . . 6 ⊢ ((φ ↔ x = y) → (φ → x = y))
4	3	ralimi 2690	. . . . 5 ⊢ (∀x ∈ A (φ ↔ x = y) → ∀x ∈ A (φ → x = y))
5	4	reximi 2722	. . . 4 ⊢ (∃y ∈ A ∀x ∈ A (φ ↔ x = y) → ∃y ∈ A ∀x ∈ A (φ → x = y))
6	2, 5	sylbi 187	. . 3 ⊢ (∃!x ∈ A φ → ∃y ∈ A ∀x ∈ A (φ → x = y))
7	1, 6	jca 518	. 2 ⊢ (∃!x ∈ A φ → (∃x ∈ A φ ∧ ∃y ∈ A ∀x ∈ A (φ → x = y)))
8		rexex 2674	. . . 4 ⊢ (∃y ∈ A ∀x ∈ A (φ → x = y) → ∃y∀x ∈ A (φ → x = y))
9	8	anim2i 552	. . 3 ⊢ ((∃x ∈ A φ ∧ ∃y ∈ A ∀x ∈ A (φ → x = y)) → (∃x ∈ A φ ∧ ∃y∀x ∈ A (φ → x = y)))
10		nfv 1619	. . . . 5 ⊢ Ⅎy(x ∈ A ∧ φ)
11	10	eu3 2230	. . . 4 ⊢ (∃!x(x ∈ A ∧ φ) ↔ (∃x(x ∈ A ∧ φ) ∧ ∃y∀x((x ∈ A ∧ φ) → x = y)))
12		df-reu 2622	. . . 4 ⊢ (∃!x ∈ A φ ↔ ∃!x(x ∈ A ∧ φ))
13		df-rex 2621	. . . . 5 ⊢ (∃x ∈ A φ ↔ ∃x(x ∈ A ∧ φ))
14		df-ral 2620	. . . . . . 7 ⊢ (∀x ∈ A (φ → x = y) ↔ ∀x(x ∈ A → (φ → x = y)))
15		impexp 433	. . . . . . . 8 ⊢ (((x ∈ A ∧ φ) → x = y) ↔ (x ∈ A → (φ → x = y)))
16	15	albii 1566	. . . . . . 7 ⊢ (∀x((x ∈ A ∧ φ) → x = y) ↔ ∀x(x ∈ A → (φ → x = y)))
17	14, 16	bitr4i 243	. . . . . 6 ⊢ (∀x ∈ A (φ → x = y) ↔ ∀x((x ∈ A ∧ φ) → x = y))
18	17	exbii 1582	. . . . 5 ⊢ (∃y∀x ∈ A (φ → x = y) ↔ ∃y∀x((x ∈ A ∧ φ) → x = y))
19	13, 18	anbi12i 678	. . . 4 ⊢ ((∃x ∈ A φ ∧ ∃y∀x ∈ A (φ → x = y)) ↔ (∃x(x ∈ A ∧ φ) ∧ ∃y∀x((x ∈ A ∧ φ) → x = y)))
20	11, 12, 19	3bitr4i 268	. . 3 ⊢ (∃!x ∈ A φ ↔ (∃x ∈ A φ ∧ ∃y∀x ∈ A (φ → x = y)))
21	9, 20	sylibr 203	. 2 ⊢ ((∃x ∈ A φ ∧ ∃y ∈ A ∀x ∈ A (φ → x = y)) → ∃!x ∈ A φ)
22	7, 21	impbii 180	1 ⊢ (∃!x ∈ A φ ↔ (∃x ∈ A φ ∧ ∃y ∈ A ∀x ∈ A (φ → x = y)))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ 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-mo 2209  df-cleq 2346  df-clel 2349  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623

This theorem is referenced by:  reu7  3032