Theorem cbviota 5554

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 1763	. . . . . 6 |- F/ z ( ph <-> x = w )
2		nfs1v 2268	. . . . . . 7 |- F/ x [ z / x ] ph
3		nfv 1763	. . . . . . 7 |- F/ x z = w
4	2, 3	nfbi 2019	. . . . . 6 |- F/ x ( [ z / x ] ph <-> z = w )
5		sbequ12 2085	. . . . . . 7 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
6		equequ1 1869	. . . . . . 7 |- ( x = z -> ( x = w <-> z = w ) )
7	5, 6	bibi12d 323	. . . . . 6 |- ( x = z -> ( ( ph <-> x = w ) <-> ( [ z / x ] ph <-> z = w ) ) )
8	1, 4, 7	cbval 2116	. . . . 5 |- ( A. x ( ph <-> x = w ) <-> A. z ( [ z / x ] ph <-> z = w ) )
9		cbviota.2	. . . . . . . 8 |- F/ y ph
10	9	nfsb 2271	. . . . . . 7 |- F/ y [ z / x ] ph
11		nfv 1763	. . . . . . 7 |- F/ y z = w
12	10, 11	nfbi 2019	. . . . . 6 |- F/ y ( [ z / x ] ph <-> z = w )
13		nfv 1763	. . . . . 6 |- F/ z ( ps <-> y = w )
14		sbequ 2207	. . . . . . . 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 2239	. . . . . . . 8 |- ( [ y / x ] ph <-> ps )
18	14, 17	syl6bb 265	. . . . . . 7 |- ( z = y -> ( [ z / x ] ph <-> ps ) )
19		equequ1 1869	. . . . . . 7 |- ( z = y -> ( z = w <-> y = w ) )
20	18, 19	bibi12d 323	. . . . . 6 |- ( z = y -> ( ( [ z / x ] ph <-> z = w ) <-> ( ps <-> y = w ) ) )
21	12, 13, 20	cbval 2116	. . . . 5 |- ( A. z ( [ z / x ] ph <-> z = w ) <-> A. y ( ps <-> y = w ) )
22	8, 21	bitri 253	. . . 4 |- ( A. x ( ph <-> x = w ) <-> A. y ( ps <-> y = w ) )
23	22	abbii 2569	. . 3 |- { w | A. x ( ph <-> x = w ) } = { w | A. y ( ps <-> y = w ) }
24	23	unieqi 4210	. 2 |- U. { w | A. x ( ph <-> x = w ) } = U. { w | A. y ( ps <-> y = w ) }
25		dfiota2 5550	. 2 |- ( iota x ph ) = U. { w | A. x ( ph <-> x = w ) }
26		dfiota2 5550	. 2 |- ( iota y ps ) = U. { w | A. y ( ps <-> y = w ) }
27	24, 25, 26	3eqtr4i 2485	1 |- ( iota x ph ) = ( iota y ps )

Syntax hints:   -> wi 4   <-> wb 188   A.wal 1444   = wceq 1446   F/wnf 1669   [wsb 1799   {cab 2439   U.cuni 4201   iotacio 5547

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-rex 2745  df-sn 3971  df-uni 4202  df-iota 5549

This theorem is referenced by:  cbviotav  5555  fvopab5  5979  cbvriota  6267