Theorem cbviota 5158

Description: Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.)

Hypotheses

Ref	Expression
cbviota.1	|- ( x = y -> ( ph <-> ps ) )
cbviota.2	|- F/ y ph
cbviota.3	|- F/ x ps

Assertion

Ref	Expression
cbviota	|- ( iota x ph ) = ( iota y ps )

Proof of Theorem cbviota

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

Step	Hyp	Ref	Expression
1		nfv 1516	. . . . . 6 |- F/ z ( ph <-> x = w )
2		nfs1v 1927	. . . . . . 7 |- F/ x [ z / x ] ph
3		nfv 1516	. . . . . . 7 |- F/ x z = w
4	2, 3	nfbi 1577	. . . . . 6 |- F/ x ( [ z / x ] ph <-> z = w )
5		sbequ12 1759	. . . . . . 7 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
6		equequ1 1700	. . . . . . 7 |- ( x = z -> ( x = w <-> z = w ) )
7	5, 6	bibi12d 234	. . . . . 6 |- ( x = z -> ( ( ph <-> x = w ) <-> ( [ z / x ] ph <-> z = w ) ) )
8	1, 4, 7	cbval 1742	. . . . 5 |- ( A. x ( ph <-> x = w ) <-> A. z ( [ z / x ] ph <-> z = w ) )
9		cbviota.2	. . . . . . . 8 |- F/ y ph
10	9	nfsb 1934	. . . . . . 7 |- F/ y [ z / x ] ph
11		nfv 1516	. . . . . . 7 |- F/ y z = w
12	10, 11	nfbi 1577	. . . . . 6 |- F/ y ( [ z / x ] ph <-> z = w )
13		nfv 1516	. . . . . 6 |- F/ z ( ps <-> y = w )
14		sbequ 1828	. . . . . . . 8 |- ( z = y -> ( [ z / x ] ph <-> [ y / x ] ph ) )
15		cbviota.3	. . . . . . . . 9 |- F/ x ps
16		cbviota.1	. . . . . . . . 9 |- ( x = y -> ( ph <-> ps ) )
17	15, 16	sbie 1779	. . . . . . . 8 |- ( [ y / x ] ph <-> ps )
18	14, 17	bitrdi 195	. . . . . . 7 |- ( z = y -> ( [ z / x ] ph <-> ps ) )
19		equequ1 1700	. . . . . . 7 |- ( z = y -> ( z = w <-> y = w ) )
20	18, 19	bibi12d 234	. . . . . 6 |- ( z = y -> ( ( [ z / x ] ph <-> z = w ) <-> ( ps <-> y = w ) ) )
21	12, 13, 20	cbval 1742	. . . . 5 |- ( A. z ( [ z / x ] ph <-> z = w ) <-> A. y ( ps <-> y = w ) )
22	8, 21	bitri 183	. . . 4 |- ( A. x ( ph <-> x = w ) <-> A. y ( ps <-> y = w ) )
23	22	abbii 2282	. . 3 |- { w | A. x ( ph <-> x = w ) } = { w | A. y ( ps <-> y = w ) }
24	23	unieqi 3799	. 2 |- U. { w | A. x ( ph <-> x = w ) } = U. { w | A. y ( ps <-> y = w ) }
25		dfiota2 5154	. 2 |- ( iota x ph ) = U. { w | A. x ( ph <-> x = w ) }
26		dfiota2 5154	. 2 |- ( iota y ps ) = U. { w | A. y ( ps <-> y = w ) }
27	24, 25, 26	3eqtr4i 2196	1 |- ( iota x ph ) = ( iota y ps )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1341   = wceq 1343   F/wnf 1448   [wsb 1750   {cab 2151   U.cuni 3789   iotacio 5151

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-sn 3582  df-uni 3790  df-iota 5153

This theorem is referenced by:  cbviotav  5159  cbvriota  5808