Theorem sb8iota 5169

Description: Variable substitution in description binder. Compare sb8eu 2033. (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 1522	. . . . . 6 |- F/ w ( ph <-> x = z )
2	1	sb8 1850	. . . . 5 |- ( A. x ( ph <-> x = z ) <-> A. w [ w / x ] ( ph <-> x = z ) )
3		sbbi 1953	. . . . . . 7 |- ( [ w / x ] ( ph <-> x = z ) <-> ( [ w / x ] ph <-> [ w / x ] x = z ) )
4		sb8iota.1	. . . . . . . . 9 |- F/ y ph
5	4	nfsb 1940	. . . . . . . 8 |- F/ y [ w / x ] ph
6		equsb3 1945	. . . . . . . . 9 |- ( [ w / x ] x = z <-> w = z )
7		nfv 1522	. . . . . . . . 9 |- F/ y w = z
8	6, 7	nfxfr 1468	. . . . . . . 8 |- F/ y [ w / x ] x = z
9	5, 8	nfbi 1583	. . . . . . 7 |- F/ y ( [ w / x ] ph <-> [ w / x ] x = z )
10	3, 9	nfxfr 1468	. . . . . 6 |- F/ y [ w / x ] ( ph <-> x = z )
11		nfv 1522	. . . . . 6 |- F/ w [ y / x ] ( ph <-> x = z )
12		sbequ 1834	. . . . . 6 |- ( w = y -> ( [ w / x ] ( ph <-> x = z ) <-> [ y / x ] ( ph <-> x = z ) ) )
13	10, 11, 12	cbval 1748	. . . . 5 |- ( A. w [ w / x ] ( ph <-> x = z ) <-> A. y [ y / x ] ( ph <-> x = z ) )
14		equsb3 1945	. . . . . . 7 |- ( [ y / x ] x = z <-> y = z )
15	14	sblbis 1954	. . . . . 6 |- ( [ y / x ] ( ph <-> x = z ) <-> ( [ y / x ] ph <-> y = z ) )
16	15	albii 1464	. . . . 5 |- ( A. y [ y / x ] ( ph <-> x = z ) <-> A. y ( [ y / x ] ph <-> y = z ) )
17	2, 13, 16	3bitri 205	. . . 4 |- ( A. x ( ph <-> x = z ) <-> A. y ( [ y / x ] ph <-> y = z ) )
18	17	abbii 2287	. . 3 |- { z | A. x ( ph <-> x = z ) } = { z | A. y ( [ y / x ] ph <-> y = z ) }
19	18	unieqi 3807	. 2 |- U. { z | A. x ( ph <-> x = z ) } = U. { z | A. y ( [ y / x ] ph <-> y = z ) }
20		dfiota2 5163	. 2 |- ( iota x ph ) = U. { z | A. x ( ph <-> x = z ) }
21		dfiota2 5163	. 2 |- ( iota y [ y / x ] ph ) = U. { z | A. y ( [ y / x ] ph <-> y = z ) }
22	19, 20, 21	3eqtr4i 2202	1 |- ( iota x ph ) = ( iota y [ y / x ] ph )

Syntax hints:   <-> wb 104   A.wal 1347   = wceq 1349   F/wnf 1454   [wsb 1756   {cab 2157   U.cuni 3797   iotacio 5160

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 705  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-10 1499  ax-11 1500  ax-i12 1501  ax-bndl 1503  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-ext 2153

This theorem depends on definitions:  df-bi 116  df-tru 1352  df-nf 1455  df-sb 1757  df-clab 2158  df-cleq 2164  df-clel 2167  df-nfc 2302  df-rex 2455  df-sn 3590  df-uni 3798  df-iota 5162

This theorem is referenced by: (None)