Theorem dfiota2 5509

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 5508	. 2 |- ( iota x ph ) = U. { y | { x | ph } = { y } }
2		df-sn 3942	. . . . . 6 |- { y } = { x | x = y }
3	2	eqeq2i 2440	. . . . 5 |- ( { x | ph } = { y } <-> { x | ph } = { x | x = y } )
4		abbi 2542	. . . . 5 |- ( A. x ( ph <-> x = y ) <-> { x | ph } = { x | x = y } )
5	3, 4	bitr4i 255	. . . 4 |- ( { x | ph } = { y } <-> A. x ( ph <-> x = y ) )
6	5	abbii 2544	. . 3 |- { y | { x | ph } = { y } } = { y | A. x ( ph <-> x = y ) }
7	6	unieqi 4171	. 2 |- U. { y | { x | ph } = { y } } = U. { y | A. x ( ph <-> x = y ) }
8	1, 7	eqtri 2450	1 |- ( iota x ph ) = U. { y | A. x ( ph <-> x = y ) }

Syntax hints:   <-> wb 187   A.wal 1435   = wceq 1437   {cab 2414   {csn 3941   U.cuni 4162   iotacio 5506

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408

This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-rex 2720  df-sn 3942  df-uni 4163  df-iota 5508

This theorem is referenced by:  nfiota1  5510  nfiotad  5511  cbviota  5513  sb8iota  5515  iotaval  5519  iotanul  5523  fv2  5820