Theorem dfiota2 5338

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 5337	. 2 |- ( iota x ph ) = U. { y | { x | ph } = { y } }
2		df-sn 3715	. . . . . 6 |- { y } = { x | x = y }
3	2	eqeq2i 2249	. . . . 5 |- ( { x | ph } = { y } <-> { x | ph } = { x | x = y } )
4		abbibcom 2352	. . . . 5 |- ( A. x ( ph <-> x = y ) <-> { x | ph } = { x | x = y } )
5	3, 4	bitr4i 187	. . . 4 |- ( { x | ph } = { y } <-> A. x ( ph <-> x = y ) )
6	5	abbii 2354	. . 3 |- { y | { x | ph } = { y } } = { y | A. x ( ph <-> x = y ) }
7	6	unieqi 3945	. 2 |- U. { y | { x | ph } = { y } } = U. { y | A. x ( ph <-> x = y ) }
8	1, 7	eqtri 2259	1 |- ( iota x ph ) = U. { y | A. x ( ph <-> x = y ) }

This proof depends on syntax axioms:   <-> wb 105   A.wal 1400   = wceq 1402   {cab 2224   {csn 3709   U.cuni 3935   iotacio 5335

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-sn 3715  df-uni 3936  df-iota 5337

This theorem is used by:  nfiota1  5339  nfiotadw  5340  cbviota  5342  sb8iota  5345  iotaval  5349  iotanul  5353  eliota  5365  fv2  5690