Theorem dfiota2 4341

Description: Alternate definition for descriptions. Definition 8.18 in [Quine] p. 56. (Contributed by Andrew Salmon, 30-Jun-2011.)

Assertion

Ref	Expression
dfiota2	⊢ (℩xφ) = ∪{y ∣ ∀x(φ ↔ x = y)}

Distinct variable groups:   x,y   φ,y

Allowed substitution hint:   φ(x)

Proof of Theorem dfiota2

Step	Hyp	Ref	Expression
1		df-iota 4340	. 2 ⊢ (℩xφ) = ∪{y ∣ {x ∣ φ} = {y}}
2		df-sn 3742	. . . . . 6 ⊢ {y} = {x ∣ x = y}
3	2	eqeq2i 2363	. . . . 5 ⊢ ({x ∣ φ} = {y} ↔ {x ∣ φ} = {x ∣ x = y})
4		abbib 2464	. . . . 5 ⊢ ({x ∣ φ} = {x ∣ x = y} ↔ ∀x(φ ↔ x = y))
5	3, 4	bitri 240	. . . 4 ⊢ ({x ∣ φ} = {y} ↔ ∀x(φ ↔ x = y))
6	5	abbii 2466	. . 3 ⊢ {y ∣ {x ∣ φ} = {y}} = {y ∣ ∀x(φ ↔ x = y)}
7	6	unieqi 3902	. 2 ⊢ ∪{y ∣ {x ∣ φ} = {y}} = ∪{y ∣ ∀x(φ ↔ x = y)}
8	1, 7	eqtri 2373	1 ⊢ (℩xφ) = ∪{y ∣ ∀x(φ ↔ x = y)}

Syntax hints:   ↔ wb 176  ∀wal 1540   = wceq 1642  {cab 2339  {csn 3738  ∪cuni 3892  ℩cio 4338

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-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-rex 2621  df-sn 3742  df-uni 3893  df-iota 4340

This theorem is referenced by:  nfiota1  4342  nfiotad  4343  cbviota  4345  sb8iota  4347  iotaval  4351  iotanul  4355  eqtfinrelk  4487  fv2  5325  tcfnex  6245