Theorem dfiota2 5535

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

Assertion

Ref	Expression
dfiota2	|- ( iota x ph ) = U. { y | A. x ( ph <-> x = y ) }

Distinct variable groups:   x, y   ph, y

Allowed substitution hint:   ph( x)

Proof of Theorem dfiota2

Step	Hyp	Ref	Expression
1		df-iota 5534	. 2 |- ( iota x ph ) = U. { y | { x | ph } = { y } }
2		df-sn 4017	. . . . . 6 |- { y } = { x | x = y }
3	2	eqeq2i 2472	. . . . 5 |- ( { x | ph } = { y } <-> { x | ph } = { x | x = y } )
4		abbi 2585	. . . . 5 |- ( A. x ( ph <-> x = y ) <-> { x | ph } = { x | x = y } )
5	3, 4	bitr4i 252	. . . 4 |- ( { x | ph } = { y } <-> A. x ( ph <-> x = y ) )
6	5	abbii 2588	. . 3 |- { y | { x | ph } = { y } } = { y | A. x ( ph <-> x = y ) }
7	6	unieqi 4244	. 2 |- U. { y | { x | ph } = { y } } = U. { y | A. x ( ph <-> x = y ) }
8	1, 7	eqtri 2483	1 |- ( iota x ph ) = U. { y | A. x ( ph <-> x = y ) }

Syntax hints:   <-> wb 184   A.wal 1396   = wceq 1398   {cab 2439   {csn 4016   U.cuni 4235   iotacio 5532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-rex 2810  df-sn 4017  df-uni 4236  df-iota 5534

This theorem is referenced by:  nfiota1  5536  nfiotad  5537  cbviota  5539  sb8iota  5541  iotaval  5545  iotanul  5549  fv2  5843