Theorem cbvriota 6268

Description: Change bound variable in a restricted description binder. (Contributed by NM, 18-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypotheses

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

Assertion

Ref	Expression
cbvriota	|- ( iota_ x e. A ph ) = ( iota_ y e. A ps )

Distinct variable groups:   x, A   y, A

Allowed substitution hints:   ph( x, y)   ps( x, y)

Proof of Theorem cbvriota

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2529	. . . . 5 |- ( x = z -> ( x e. A <-> z e. A ) )
2		sbequ12 1993	. . . . 5 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
3	1, 2	anbi12d 710	. . . 4 |- ( x = z -> ( ( x e. A /\ ph ) <-> ( z e. A /\ [ z / x ] ph ) ) )
4		nfv 1708	. . . 4 |- F/ z ( x e. A /\ ph )
5		nfv 1708	. . . . 5 |- F/ x z e. A
6		nfs1v 2182	. . . . 5 |- F/ x [ z / x ] ph
7	5, 6	nfan 1929	. . . 4 |- F/ x ( z e. A /\ [ z / x ] ph )
8	3, 4, 7	cbviota 5562	. . 3 |- ( iota x ( x e. A /\ ph ) ) = ( iota z ( z e. A /\ [ z / x ] ph ) )
9		eleq1 2529	. . . . 5 |- ( z = y -> ( z e. A <-> y e. A ) )
10		sbequ 2118	. . . . . 6 |- ( z = y -> ( [ z / x ] ph <-> [ y / x ] ph ) )
11		cbvriota.2	. . . . . . 7 |- F/ x ps
12		cbvriota.3	. . . . . . 7 |- ( x = y -> ( ph <-> ps ) )
13	11, 12	sbie 2150	. . . . . 6 |- ( [ y / x ] ph <-> ps )
14	10, 13	syl6bb 261	. . . . 5 |- ( z = y -> ( [ z / x ] ph <-> ps ) )
15	9, 14	anbi12d 710	. . . 4 |- ( z = y -> ( ( z e. A /\ [ z / x ] ph ) <-> ( y e. A /\ ps ) ) )
16		nfv 1708	. . . . 5 |- F/ y z e. A
17		cbvriota.1	. . . . . 6 |- F/ y ph
18	17	nfsb 2185	. . . . 5 |- F/ y [ z / x ] ph
19	16, 18	nfan 1929	. . . 4 |- F/ y ( z e. A /\ [ z / x ] ph )
20		nfv 1708	. . . 4 |- F/ z ( y e. A /\ ps )
21	15, 19, 20	cbviota 5562	. . 3 |- ( iota z ( z e. A /\ [ z / x ] ph ) ) = ( iota y ( y e. A /\ ps ) )
22	8, 21	eqtri 2486	. 2 |- ( iota x ( x e. A /\ ph ) ) = ( iota y ( y e. A /\ ps ) )
23		df-riota 6258	. 2 |- ( iota_ x e. A ph ) = ( iota x ( x e. A /\ ph ) )
24		df-riota 6258	. 2 |- ( iota_ y e. A ps ) = ( iota y ( y e. A /\ ps ) )
25	22, 23, 24	3eqtr4i 2496	1 |- ( iota_ x e. A ph ) = ( iota_ y e. A ps )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1395   F/wnf 1617   [wsb 1740   e. wcel 1819   iotacio 5555   iota_crio 6257

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  df-riota 6258

This theorem is referenced by:  cbvriotav  6269