Theorem iotaval 5240

Description: Theorem 8.19 in [Quine] p. 57. This theorem is the fundamental property of iota. (Contributed by Andrew Salmon, 11-Jul-2011.)

Assertion

Ref	Expression
iotaval	|- ( A. x ( ph <-> x = y ) -> ( iota x ph ) = y )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem iotaval

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfiota2 5230	. 2 |- ( iota x ph ) = U. { z | A. x ( ph <-> x = z ) }
2		vex 2774	. . . . . . 7 |- y e. _V
3		sbeqalb 3054	. . . . . . . 8 |- ( y e. _V -> ( ( A. x ( ph <-> x = y ) /\ A. x ( ph <-> x = z ) ) -> y = z ) )
4		equcomi 1726	. . . . . . . 8 |- ( y = z -> z = y )
5	3, 4	syl6 33	. . . . . . 7 |- ( y e. _V -> ( ( A. x ( ph <-> x = y ) /\ A. x ( ph <-> x = z ) ) -> z = y ) )
6	2, 5	ax-mp 5	. . . . . 6 |- ( ( A. x ( ph <-> x = y ) /\ A. x ( ph <-> x = z ) ) -> z = y )
7	6	ex 115	. . . . 5 |- ( A. x ( ph <-> x = y ) -> ( A. x ( ph <-> x = z ) -> z = y ) )
8		equequ2 1735	. . . . . . . . . 10 |- ( y = z -> ( x = y <-> x = z ) )
9	8	equcoms 1730	. . . . . . . . 9 |- ( z = y -> ( x = y <-> x = z ) )
10	9	bibi2d 232	. . . . . . . 8 |- ( z = y -> ( ( ph <-> x = y ) <-> ( ph <-> x = z ) ) )
11	10	biimpd 144	. . . . . . 7 |- ( z = y -> ( ( ph <-> x = y ) -> ( ph <-> x = z ) ) )
12	11	alimdv 1901	. . . . . 6 |- ( z = y -> ( A. x ( ph <-> x = y ) -> A. x ( ph <-> x = z ) ) )
13	12	com12 30	. . . . 5 |- ( A. x ( ph <-> x = y ) -> ( z = y -> A. x ( ph <-> x = z ) ) )
14	7, 13	impbid 129	. . . 4 |- ( A. x ( ph <-> x = y ) -> ( A. x ( ph <-> x = z ) <-> z = y ) )
15	14	alrimiv 1896	. . 3 |- ( A. x ( ph <-> x = y ) -> A. z ( A. x ( ph <-> x = z ) <-> z = y ) )
16		uniabio 5239	. . 3 |- ( A. z ( A. x ( ph <-> x = z ) <-> z = y ) -> U. { z | A. x ( ph <-> x = z ) } = y )
17	15, 16	syl 14	. 2 |- ( A. x ( ph <-> x = y ) -> U. { z | A. x ( ph <-> x = z ) } = y )
18	1, 17	eqtrid 2249	1 |- ( A. x ( ph <-> x = y ) -> ( iota x ph ) = y )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1370   = wceq 1372   e. wcel 2175   {cab 2190   _Vcvv 2771   U.cuni 3849   iotacio 5227

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-rex 2489  df-v 2773  df-sbc 2998  df-un 3169  df-sn 3638  df-pr 3639  df-uni 3850  df-iota 5229

This theorem is referenced by:  iotauni  5241  iota1  5243  euiotaex  5245  iota4  5248  iota5  5250