Theorem sb8iota 5286

Description: Variable substitution in description binder. Compare sb8eu 2090. (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 1574	. . . . . 6 |- F/ w ( ph <-> x = z )
2	1	sb8 1902	. . . . 5 |- ( A. x ( ph <-> x = z ) <-> A. w [ w / x ] ( ph <-> x = z ) )
3		sbbi 2010	. . . . . . 7 |- ( [ w / x ] ( ph <-> x = z ) <-> ( [ w / x ] ph <-> [ w / x ] x = z ) )
4		sb8iota.1	. . . . . . . . 9 |- F/ y ph
5	4	nfsb 1997	. . . . . . . 8 |- F/ y [ w / x ] ph
6		equsb3 2002	. . . . . . . . 9 |- ( [ w / x ] x = z <-> w = z )
7		nfv 1574	. . . . . . . . 9 |- F/ y w = z
8	6, 7	nfxfr 1520	. . . . . . . 8 |- F/ y [ w / x ] x = z
9	5, 8	nfbi 1635	. . . . . . 7 |- F/ y ( [ w / x ] ph <-> [ w / x ] x = z )
10	3, 9	nfxfr 1520	. . . . . 6 |- F/ y [ w / x ] ( ph <-> x = z )
11		nfv 1574	. . . . . 6 |- F/ w [ y / x ] ( ph <-> x = z )
12		sbequ 1886	. . . . . 6 |- ( w = y -> ( [ w / x ] ( ph <-> x = z ) <-> [ y / x ] ( ph <-> x = z ) ) )
13	10, 11, 12	cbval 1800	. . . . 5 |- ( A. w [ w / x ] ( ph <-> x = z ) <-> A. y [ y / x ] ( ph <-> x = z ) )
14		equsb3 2002	. . . . . . 7 |- ( [ y / x ] x = z <-> y = z )
15	14	sblbis 2011	. . . . . 6 |- ( [ y / x ] ( ph <-> x = z ) <-> ( [ y / x ] ph <-> y = z ) )
16	15	albii 1516	. . . . 5 |- ( A. y [ y / x ] ( ph <-> x = z ) <-> A. y ( [ y / x ] ph <-> y = z ) )
17	2, 13, 16	3bitri 206	. . . 4 |- ( A. x ( ph <-> x = z ) <-> A. y ( [ y / x ] ph <-> y = z ) )
18	17	abbii 2345	. . 3 |- { z | A. x ( ph <-> x = z ) } = { z | A. y ( [ y / x ] ph <-> y = z ) }
19	18	unieqi 3898	. 2 |- U. { z | A. x ( ph <-> x = z ) } = U. { z | A. y ( [ y / x ] ph <-> y = z ) }
20		dfiota2 5279	. 2 |- ( iota x ph ) = U. { z | A. x ( ph <-> x = z ) }
21		dfiota2 5279	. 2 |- ( iota y [ y / x ] ph ) = U. { z | A. y ( [ y / x ] ph <-> y = z ) }
22	19, 20, 21	3eqtr4i 2260	1 |- ( iota x ph ) = ( iota y [ y / x ] ph )

Syntax hints:   <-> wb 105   A.wal 1393   = wceq 1395   F/wnf 1506   [wsb 1808   {cab 2215   U.cuni 3888   iotacio 5276

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-sn 3672  df-uni 3889  df-iota 5278

This theorem is referenced by: (None)