Theorem iotaval 5099

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 5089	. 2 |- ( iota x ph ) = U. { z | A. x ( ph <-> x = z ) }
2		vex 2689	. . . . . . 7 |- y e. _V
3		sbeqalb 2965	. . . . . . . 8 |- ( y e. _V -> ( ( A. x ( ph <-> x = y ) /\ A. x ( ph <-> x = z ) ) -> y = z ) )
4		equcomi 1680	. . . . . . . 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 114	. . . . 5 |- ( A. x ( ph <-> x = y ) -> ( A. x ( ph <-> x = z ) -> z = y ) )
8		equequ2 1689	. . . . . . . . . 10 |- ( y = z -> ( x = y <-> x = z ) )
9	8	equcoms 1684	. . . . . . . . 9 |- ( z = y -> ( x = y <-> x = z ) )
10	9	bibi2d 231	. . . . . . . 8 |- ( z = y -> ( ( ph <-> x = y ) <-> ( ph <-> x = z ) ) )
11	10	biimpd 143	. . . . . . 7 |- ( z = y -> ( ( ph <-> x = y ) -> ( ph <-> x = z ) ) )
12	11	alimdv 1851	. . . . . 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 128	. . . 4 |- ( A. x ( ph <-> x = y ) -> ( A. x ( ph <-> x = z ) <-> z = y ) )
15	14	alrimiv 1846	. . 3 |- ( A. x ( ph <-> x = y ) -> A. z ( A. x ( ph <-> x = z ) <-> z = y ) )
16		uniabio 5098	. . 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	syl5eq 2184	1 |- ( A. x ( ph <-> x = y ) -> ( iota x ph ) = y )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1329   = wceq 1331   e. wcel 1480   {cab 2125   _Vcvv 2686   U.cuni 3736   iotacio 5086

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-sn 3533  df-pr 3534  df-uni 3737  df-iota 5088

This theorem is referenced by:  iotauni  5100  iota1  5102  euiotaex  5104  iota4  5106  iota5  5108