Theorem iotaval 5862

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 5852	. 2 |- ( iota x ph ) = U. { z | A. x ( ph <-> x = z ) }
2		vex 3203	. . . . . . . 8 |- y e. _V
3		sbeqalb 3488	. . . . . . . 8 |- ( y e. _V -> ( ( A. x ( ph <-> x = y ) /\ A. x ( ph <-> x = z ) ) -> y = z ) )
4	2, 3	ax-mp 5	. . . . . . 7 |- ( ( A. x ( ph <-> x = y ) /\ A. x ( ph <-> x = z ) ) -> y = z )
5	4	ex 450	. . . . . 6 |- ( A. x ( ph <-> x = y ) -> ( A. x ( ph <-> x = z ) -> y = z ) )
6		equequ2 1953	. . . . . . . . . 10 |- ( y = z -> ( x = y <-> x = z ) )
7	6	bibi2d 332	. . . . . . . . 9 |- ( y = z -> ( ( ph <-> x = y ) <-> ( ph <-> x = z ) ) )
8	7	biimpd 219	. . . . . . . 8 |- ( y = z -> ( ( ph <-> x = y ) -> ( ph <-> x = z ) ) )
9	8	alimdv 1845	. . . . . . 7 |- ( y = z -> ( A. x ( ph <-> x = y ) -> A. x ( ph <-> x = z ) ) )
10	9	com12 32	. . . . . 6 |- ( A. x ( ph <-> x = y ) -> ( y = z -> A. x ( ph <-> x = z ) ) )
11	5, 10	impbid 202	. . . . 5 |- ( A. x ( ph <-> x = y ) -> ( A. x ( ph <-> x = z ) <-> y = z ) )
12		equcom 1945	. . . . 5 |- ( y = z <-> z = y )
13	11, 12	syl6bb 276	. . . 4 |- ( A. x ( ph <-> x = y ) -> ( A. x ( ph <-> x = z ) <-> z = y ) )
14	13	alrimiv 1855	. . 3 |- ( A. x ( ph <-> x = y ) -> A. z ( A. x ( ph <-> x = z ) <-> z = y ) )
15		uniabio 5861	. . 3 |- ( A. z ( A. x ( ph <-> x = z ) <-> z = y ) -> U. { z | A. x ( ph <-> x = z ) } = y )
16	14, 15	syl 17	. 2 |- ( A. x ( ph <-> x = y ) -> U. { z | A. x ( ph <-> x = z ) } = y )
17	1, 16	syl5eq 2668	1 |- ( A. x ( ph <-> x = y ) -> ( iota x ph ) = y )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   {cab 2608   _Vcvv 3200   U.cuni 4436   iotacio 5849

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-v 3202  df-sbc 3436  df-un 3579  df-sn 4178  df-pr 4180  df-uni 4437  df-iota 5851

This theorem is referenced by:  iotauni  5863  iota1  5865  iotaex  5868  iota4  5869  iota5  5871  iota5f  31606  iotain  38618  iotaexeu  38619  iotasbc  38620  iotaequ  38630  iotavalb  38631  pm14.24  38633  sbiota1  38635