Theorem sb8iota 4894

Description: Variable substitution in description binder. Compare sb8eu 1954. (Contributed by NM, 18-Mar-2013.)

Hypothesis

Ref	Expression
sb8iota.1	|- F/ y ph

Assertion

Ref	Expression
sb8iota	|- ( iota x ph ) = ( iota y [ y / x ] ph )

Proof of Theorem sb8iota

Dummy variables z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1461	. . . . . 6 |- F/ w ( ph <-> x = z )
2	1	sb8 1777	. . . . 5 |- ( A. x ( ph <-> x = z ) <-> A. w [ w / x ] ( ph <-> x = z ) )
3		sbbi 1874	. . . . . . 7 |- ( [ w / x ] ( ph <-> x = z ) <-> ( [ w / x ] ph <-> [ w / x ] x = z ) )
4		sb8iota.1	. . . . . . . . 9 |- F/ y ph
5	4	nfsb 1863	. . . . . . . 8 |- F/ y [ w / x ] ph
6		equsb3 1866	. . . . . . . . 9 |- ( [ w / x ] x = z <-> w = z )
7		nfv 1461	. . . . . . . . 9 |- F/ y w = z
8	6, 7	nfxfr 1403	. . . . . . . 8 |- F/ y [ w / x ] x = z
9	5, 8	nfbi 1521	. . . . . . 7 |- F/ y ( [ w / x ] ph <-> [ w / x ] x = z )
10	3, 9	nfxfr 1403	. . . . . 6 |- F/ y [ w / x ] ( ph <-> x = z )
11		nfv 1461	. . . . . 6 |- F/ w [ y / x ] ( ph <-> x = z )
12		sbequ 1761	. . . . . 6 |- ( w = y -> ( [ w / x ] ( ph <-> x = z ) <-> [ y / x ] ( ph <-> x = z ) ) )
13	10, 11, 12	cbval 1677	. . . . 5 |- ( A. w [ w / x ] ( ph <-> x = z ) <-> A. y [ y / x ] ( ph <-> x = z ) )
14		equsb3 1866	. . . . . . 7 |- ( [ y / x ] x = z <-> y = z )
15	14	sblbis 1875	. . . . . 6 |- ( [ y / x ] ( ph <-> x = z ) <-> ( [ y / x ] ph <-> y = z ) )
16	15	albii 1399	. . . . 5 |- ( A. y [ y / x ] ( ph <-> x = z ) <-> A. y ( [ y / x ] ph <-> y = z ) )
17	2, 13, 16	3bitri 204	. . . 4 |- ( A. x ( ph <-> x = z ) <-> A. y ( [ y / x ] ph <-> y = z ) )
18	17	abbii 2194	. . 3 |- { z | A. x ( ph <-> x = z ) } = { z | A. y ( [ y / x ] ph <-> y = z ) }
19	18	unieqi 3611	. 2 |- U. { z | A. x ( ph <-> x = z ) } = U. { z | A. y ( [ y / x ] ph <-> y = z ) }
20		dfiota2 4888	. 2 |- ( iota x ph ) = U. { z | A. x ( ph <-> x = z ) }
21		dfiota2 4888	. 2 |- ( iota y [ y / x ] ph ) = U. { z | A. y ( [ y / x ] ph <-> y = z ) }
22	19, 20, 21	3eqtr4i 2111	1 |- ( iota x ph ) = ( iota y [ y / x ] ph )

Syntax hints:   <-> wb 103   A.wal 1282   = wceq 1284   F/wnf 1389   [wsb 1685   {cab 2067   U.cuni 3601   iotacio 4885

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354  df-sn 3404  df-uni 3602  df-iota 4887

This theorem is referenced by: (None)