Theorem cbviota

Description: Change bound variables in a description binder. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker cbviotaw when possible. (Contributed by Andrew Salmon, 1-Aug-2011) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		cbviota.1	|- ( x = y -> ( ph <-> ps ) )
2		cbviota.2	|- F/ y ph
3		cbviota.3	|- F/ x ps
4		nfv	|- F/ z ( ph <-> x = w )
5		nfs1v	|- F/ x [ z / x ] ph
6		nfv	|- F/ x z = w
7	5 6	nfbi	|- F/ x ( [ z / x ] ph <-> z = w )
8		sbequ12	|- ( x = z -> ( ph <-> [ z / x ] ph ) )
9		equequ1	|- ( x = z -> ( x = w <-> z = w ) )
10	8 9	bibi12d	|- ( x = z -> ( ( ph <-> x = w ) <-> ( [ z / x ] ph <-> z = w ) ) )
11	4 7 10	cbvalv1	|- ( A. x ( ph <-> x = w ) <-> A. z ( [ z / x ] ph <-> z = w ) )
12	2	nfsb	|- F/ y [ z / x ] ph
13		nfv	|- F/ y z = w
14	12 13	nfbi	|- F/ y ( [ z / x ] ph <-> z = w )
15		nfv	|- F/ z ( ps <-> y = w )
16		sbequ	|- ( z = y -> ( [ z / x ] ph <-> [ y / x ] ph ) )
17	3 1	sbie	|- ( [ y / x ] ph <-> ps )
18	16 17	bitrdi	|- ( z = y -> ( [ z / x ] ph <-> ps ) )
19		equequ1	|- ( z = y -> ( z = w <-> y = w ) )
20	18 19	bibi12d	|- ( z = y -> ( ( [ z / x ] ph <-> z = w ) <-> ( ps <-> y = w ) ) )
21	14 15 20	cbvalv1	|- ( A. z ( [ z / x ] ph <-> z = w ) <-> A. y ( ps <-> y = w ) )
22	11 21	bitri	|- ( A. x ( ph <-> x = w ) <-> A. y ( ps <-> y = w ) )
23	22	abbii	|- { w | A. x ( ph <-> x = w ) } = { w | A. y ( ps <-> y = w ) }
24	23	unieqi	|- U. { w | A. x ( ph <-> x = w ) } = U. { w | A. y ( ps <-> y = w ) }
25		dfiota2	|- ( iota x ph ) = U. { w | A. x ( ph <-> x = w ) }
26		dfiota2	|- ( iota y ps ) = U. { w | A. y ( ps <-> y = w ) }
27	24 25 26	3eqtr4i	|- ( iota x ph ) = ( iota y ps )