Theorem cbviota 5562

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 1708	. . . . . 6 |- F/ z ( ph <-> x = w )
2		nfs1v 2182	. . . . . . 7 |- F/ x [ z / x ] ph
3		nfv 1708	. . . . . . 7 |- F/ x z = w
4	2, 3	nfbi 1935	. . . . . 6 |- F/ x ( [ z / x ] ph <-> z = w )
5		sbequ12 1993	. . . . . . 7 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
6		equequ1 1799	. . . . . . 7 |- ( x = z -> ( x = w <-> z = w ) )
7	5, 6	bibi12d 321	. . . . . 6 |- ( x = z -> ( ( ph <-> x = w ) <-> ( [ z / x ] ph <-> z = w ) ) )
8	1, 4, 7	cbval 2022	. . . . 5 |- ( A. x ( ph <-> x = w ) <-> A. z ( [ z / x ] ph <-> z = w ) )
9		cbviota.2	. . . . . . . 8 |- F/ y ph
10	9	nfsb 2185	. . . . . . 7 |- F/ y [ z / x ] ph
11		nfv 1708	. . . . . . 7 |- F/ y z = w
12	10, 11	nfbi 1935	. . . . . 6 |- F/ y ( [ z / x ] ph <-> z = w )
13		nfv 1708	. . . . . 6 |- F/ z ( ps <-> y = w )
14		sbequ 2118	. . . . . . . 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 2150	. . . . . . . 8 |- ( [ y / x ] ph <-> ps )
18	14, 17	syl6bb 261	. . . . . . 7 |- ( z = y -> ( [ z / x ] ph <-> ps ) )
19		equequ1 1799	. . . . . . 7 |- ( z = y -> ( z = w <-> y = w ) )
20	18, 19	bibi12d 321	. . . . . 6 |- ( z = y -> ( ( [ z / x ] ph <-> z = w ) <-> ( ps <-> y = w ) ) )
21	12, 13, 20	cbval 2022	. . . . 5 |- ( A. z ( [ z / x ] ph <-> z = w ) <-> A. y ( ps <-> y = w ) )
22	8, 21	bitri 249	. . . 4 |- ( A. x ( ph <-> x = w ) <-> A. y ( ps <-> y = w ) )
23	22	abbii 2591	. . 3 |- { w | A. x ( ph <-> x = w ) } = { w | A. y ( ps <-> y = w ) }
24	23	unieqi 4260	. 2 |- U. { w | A. x ( ph <-> x = w ) } = U. { w | A. y ( ps <-> y = w ) }
25		dfiota2 5558	. 2 |- ( iota x ph ) = U. { w | A. x ( ph <-> x = w ) }
26		dfiota2 5558	. 2 |- ( iota y ps ) = U. { w | A. y ( ps <-> y = w ) }
27	24, 25, 26	3eqtr4i 2496	1 |- ( iota x ph ) = ( iota y ps )

Syntax hints:   -> wi 4   <-> wb 184   A.wal 1393   = wceq 1395   F/wnf 1617   [wsb 1740   {cab 2442   U.cuni 4251   iotacio 5555

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-rex 2813  df-sn 4033  df-uni 4252  df-iota 5557

This theorem is referenced by:  cbviotav  5563  fvopab5  5980  cbvriota  6268