Theorem sb8iota 5858

Description: Variable substitution in description binder. Compare sb8eu 2503. (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 1843	. . . . . 6 |- F/ w ( ph <-> x = z )
2	1	sb8 2424	. . . . 5 |- ( A. x ( ph <-> x = z ) <-> A. w [ w / x ] ( ph <-> x = z ) )
3		sbbi 2401	. . . . . . 7 |- ( [ w / x ] ( ph <-> x = z ) <-> ( [ w / x ] ph <-> [ w / x ] x = z ) )
4		sb8iota.1	. . . . . . . . 9 |- F/ y ph
5	4	nfsb 2440	. . . . . . . 8 |- F/ y [ w / x ] ph
6		equsb3 2432	. . . . . . . . 9 |- ( [ w / x ] x = z <-> w = z )
7		nfv 1843	. . . . . . . . 9 |- F/ y w = z
8	6, 7	nfxfr 1779	. . . . . . . 8 |- F/ y [ w / x ] x = z
9	5, 8	nfbi 1833	. . . . . . 7 |- F/ y ( [ w / x ] ph <-> [ w / x ] x = z )
10	3, 9	nfxfr 1779	. . . . . 6 |- F/ y [ w / x ] ( ph <-> x = z )
11		nfv 1843	. . . . . 6 |- F/ w [ y / x ] ( ph <-> x = z )
12		sbequ 2376	. . . . . 6 |- ( w = y -> ( [ w / x ] ( ph <-> x = z ) <-> [ y / x ] ( ph <-> x = z ) ) )
13	10, 11, 12	cbval 2271	. . . . 5 |- ( A. w [ w / x ] ( ph <-> x = z ) <-> A. y [ y / x ] ( ph <-> x = z ) )
14		equsb3 2432	. . . . . . 7 |- ( [ y / x ] x = z <-> y = z )
15	14	sblbis 2404	. . . . . 6 |- ( [ y / x ] ( ph <-> x = z ) <-> ( [ y / x ] ph <-> y = z ) )
16	15	albii 1747	. . . . 5 |- ( A. y [ y / x ] ( ph <-> x = z ) <-> A. y ( [ y / x ] ph <-> y = z ) )
17	2, 13, 16	3bitri 286	. . . 4 |- ( A. x ( ph <-> x = z ) <-> A. y ( [ y / x ] ph <-> y = z ) )
18	17	abbii 2739	. . 3 |- { z | A. x ( ph <-> x = z ) } = { z | A. y ( [ y / x ] ph <-> y = z ) }
19	18	unieqi 4445	. 2 |- U. { z | A. x ( ph <-> x = z ) } = U. { z | A. y ( [ y / x ] ph <-> y = z ) }
20		dfiota2 5852	. 2 |- ( iota x ph ) = U. { z | A. x ( ph <-> x = z ) }
21		dfiota2 5852	. 2 |- ( iota y [ y / x ] ph ) = U. { z | A. y ( [ y / x ] ph <-> y = z ) }
22	19, 20, 21	3eqtr4i 2654	1 |- ( iota x ph ) = ( iota y [ y / x ] ph )

Syntax hints:   <-> wb 196   A.wal 1481   = wceq 1483   F/wnf 1708   [wsb 1880   {cab 2608   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-sn 4178  df-uni 4437  df-iota 5851

This theorem is referenced by: (None)