Theorem sb8iota 5154

Description: Variable substitution in description binder. Compare sb8eu 2026. (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 1515	. . . . . 6 |- F/ w ( ph <-> x = z )
2	1	sb8 1843	. . . . 5 |- ( A. x ( ph <-> x = z ) <-> A. w [ w / x ] ( ph <-> x = z ) )
3		sbbi 1946	. . . . . . 7 |- ( [ w / x ] ( ph <-> x = z ) <-> ( [ w / x ] ph <-> [ w / x ] x = z ) )
4		sb8iota.1	. . . . . . . . 9 |- F/ y ph
5	4	nfsb 1933	. . . . . . . 8 |- F/ y [ w / x ] ph
6		equsb3 1938	. . . . . . . . 9 |- ( [ w / x ] x = z <-> w = z )
7		nfv 1515	. . . . . . . . 9 |- F/ y w = z
8	6, 7	nfxfr 1461	. . . . . . . 8 |- F/ y [ w / x ] x = z
9	5, 8	nfbi 1576	. . . . . . 7 |- F/ y ( [ w / x ] ph <-> [ w / x ] x = z )
10	3, 9	nfxfr 1461	. . . . . 6 |- F/ y [ w / x ] ( ph <-> x = z )
11		nfv 1515	. . . . . 6 |- F/ w [ y / x ] ( ph <-> x = z )
12		sbequ 1827	. . . . . 6 |- ( w = y -> ( [ w / x ] ( ph <-> x = z ) <-> [ y / x ] ( ph <-> x = z ) ) )
13	10, 11, 12	cbval 1741	. . . . 5 |- ( A. w [ w / x ] ( ph <-> x = z ) <-> A. y [ y / x ] ( ph <-> x = z ) )
14		equsb3 1938	. . . . . . 7 |- ( [ y / x ] x = z <-> y = z )
15	14	sblbis 1947	. . . . . 6 |- ( [ y / x ] ( ph <-> x = z ) <-> ( [ y / x ] ph <-> y = z ) )
16	15	albii 1457	. . . . 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 2280	. . 3 |- { z | A. x ( ph <-> x = z ) } = { z | A. y ( [ y / x ] ph <-> y = z ) }
19	18	unieqi 3793	. 2 |- U. { z | A. x ( ph <-> x = z ) } = U. { z | A. y ( [ y / x ] ph <-> y = z ) }
20		dfiota2 5148	. 2 |- ( iota x ph ) = U. { z | A. x ( ph <-> x = z ) }
21		dfiota2 5148	. 2 |- ( iota y [ y / x ] ph ) = U. { z | A. y ( [ y / x ] ph <-> y = z ) }
22	19, 20, 21	3eqtr4i 2195	1 |- ( iota x ph ) = ( iota y [ y / x ] ph )

Syntax hints:   <-> wb 104   A.wal 1340   = wceq 1342   F/wnf 1447   [wsb 1749   {cab 2150   U.cuni 3783   iotacio 5145

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 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-rex 2448  df-sn 3576  df-uni 3784  df-iota 5147

This theorem is referenced by: (None)