Theorem cbviota 5165

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 1521	. . . . . 6 |- F/ z ( ph <-> x = w )
2		nfs1v 1932	. . . . . . 7 |- F/ x [ z / x ] ph
3		nfv 1521	. . . . . . 7 |- F/ x z = w
4	2, 3	nfbi 1582	. . . . . 6 |- F/ x ( [ z / x ] ph <-> z = w )
5		sbequ12 1764	. . . . . . 7 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
6		equequ1 1705	. . . . . . 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 1747	. . . . 5 |- ( A. x ( ph <-> x = w ) <-> A. z ( [ z / x ] ph <-> z = w ) )
9		cbviota.2	. . . . . . . 8 |- F/ y ph
10	9	nfsb 1939	. . . . . . 7 |- F/ y [ z / x ] ph
11		nfv 1521	. . . . . . 7 |- F/ y z = w
12	10, 11	nfbi 1582	. . . . . 6 |- F/ y ( [ z / x ] ph <-> z = w )
13		nfv 1521	. . . . . 6 |- F/ z ( ps <-> y = w )
14		sbequ 1833	. . . . . . . 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 1784	. . . . . . . 8 |- ( [ y / x ] ph <-> ps )
18	14, 17	bitrdi 195	. . . . . . 7 |- ( z = y -> ( [ z / x ] ph <-> ps ) )
19		equequ1 1705	. . . . . . 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 1747	. . . . 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 2286	. . 3 |- { w | A. x ( ph <-> x = w ) } = { w | A. y ( ps <-> y = w ) }
24	23	unieqi 3806	. 2 |- U. { w | A. x ( ph <-> x = w ) } = U. { w | A. y ( ps <-> y = w ) }
25		dfiota2 5161	. 2 |- ( iota x ph ) = U. { w | A. x ( ph <-> x = w ) }
26		dfiota2 5161	. 2 |- ( iota y ps ) = U. { w | A. y ( ps <-> y = w ) }
27	24, 25, 26	3eqtr4i 2201	1 |- ( iota x ph ) = ( iota y ps )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1346   = wceq 1348   F/wnf 1453   [wsb 1755   {cab 2156   U.cuni 3796   iotacio 5158

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-sn 3589  df-uni 3797  df-iota 5160

This theorem is referenced by:  cbviotav  5166  cbvriota  5819