Theorem iotaval 4878

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 4868	. 2 |- ( iota x ph ) = U. { z | A. x ( ph <-> x = z ) }
2		vex 2560	. . . . . . 7 |- y e. _V
3		sbeqalb 2815	. . . . . . . 8 |- ( y e. _V -> ( ( A. x ( ph <-> x = y ) /\ A. x ( ph <-> x = z ) ) -> y = z ) )
4		equcomi 1592	. . . . . . . 8 |- ( y = z -> z = y )
5	3, 4	syl6 29	. . . . . . 7 |- ( y e. _V -> ( ( A. x ( ph <-> x = y ) /\ A. x ( ph <-> x = z ) ) -> z = y ) )
6	2, 5	ax-mp 7	. . . . . 6 |- ( ( A. x ( ph <-> x = y ) /\ A. x ( ph <-> x = z ) ) -> z = y )
7	6	ex 108	. . . . 5 |- ( A. x ( ph <-> x = y ) -> ( A. x ( ph <-> x = z ) -> z = y ) )
8		equequ2 1599	. . . . . . . . . 10 |- ( y = z -> ( x = y <-> x = z ) )
9	8	equcoms 1594	. . . . . . . . 9 |- ( z = y -> ( x = y <-> x = z ) )
10	9	bibi2d 221	. . . . . . . 8 |- ( z = y -> ( ( ph <-> x = y ) <-> ( ph <-> x = z ) ) )
11	10	biimpd 132	. . . . . . 7 |- ( z = y -> ( ( ph <-> x = y ) -> ( ph <-> x = z ) ) )
12	11	alimdv 1759	. . . . . 6 |- ( z = y -> ( A. x ( ph <-> x = y ) -> A. x ( ph <-> x = z ) ) )
13	12	com12 27	. . . . 5 |- ( A. x ( ph <-> x = y ) -> ( z = y -> A. x ( ph <-> x = z ) ) )
14	7, 13	impbid 120	. . . 4 |- ( A. x ( ph <-> x = y ) -> ( A. x ( ph <-> x = z ) <-> z = y ) )
15	14	alrimiv 1754	. . 3 |- ( A. x ( ph <-> x = y ) -> A. z ( A. x ( ph <-> x = z ) <-> z = y ) )
16		uniabio 4877	. . 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 2084	1 |- ( A. x ( ph <-> x = y ) -> ( iota x ph ) = y )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   A.wal 1241   = wceq 1243   e. wcel 1393   {cab 2026   _Vcvv 2557   U.cuni 3580   iotacio 4865

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-sn 3381  df-pr 3382  df-uni 3581  df-iota 4867

This theorem is referenced by:  iotauni  4879  iota1  4881  euiotaex  4883  iota4  4885  iota5  4887