Theorem euf 2210

Description: A version of the existential uniqueness definition with a hypothesis instead of a distinct variable condition. (Contributed by NM, 12-Aug-1993.)

Hypothesis

Ref	Expression
euf.1	⊢ Ⅎyφ

Assertion

Ref	Expression
euf	⊢ (∃!xφ ↔ ∃y∀x(φ ↔ x = y))

Distinct variable group:   x,y

Allowed substitution hints:   φ(x,y)

Proof of Theorem euf

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-eu 2208	. 2 ⊢ (∃!xφ ↔ ∃z∀x(φ ↔ x = z))
2		euf.1	. . . . 5 ⊢ Ⅎyφ
3		nfv 1619	. . . . 5 ⊢ Ⅎy x = z
4	2, 3	nfbi 1834	. . . 4 ⊢ Ⅎy(φ ↔ x = z)
5	4	nfal 1842	. . 3 ⊢ Ⅎy∀x(φ ↔ x = z)
6		nfv 1619	. . . . 5 ⊢ Ⅎzφ
7		nfv 1619	. . . . 5 ⊢ Ⅎz x = y
8	6, 7	nfbi 1834	. . . 4 ⊢ Ⅎz(φ ↔ x = y)
9	8	nfal 1842	. . 3 ⊢ Ⅎz∀x(φ ↔ x = y)
10		equequ2 1686	. . . . 5 ⊢ (z = y → (x = z ↔ x = y))
11	10	bibi2d 309	. . . 4 ⊢ (z = y → ((φ ↔ x = z) ↔ (φ ↔ x = y)))
12	11	albidv 1625	. . 3 ⊢ (z = y → (∀x(φ ↔ x = z) ↔ ∀x(φ ↔ x = y)))
13	5, 9, 12	cbvex 1985	. 2 ⊢ (∃z∀x(φ ↔ x = z) ↔ ∃y∀x(φ ↔ x = y))
14	1, 13	bitri 240	1 ⊢ (∃!xφ ↔ ∃y∀x(φ ↔ x = y))

Syntax hints:   ↔ wb 176  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544   = wceq 1642  ∃!weu 2204

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

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-eu 2208

This theorem is referenced by:  eu1  2225  eumo0  2228