Theorem cbvriota 5787

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 2220	. . . . 5 |- ( x = z -> ( x e. A <-> z e. A ) )
2		sbequ12 1751	. . . . 5 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
3	1, 2	anbi12d 465	. . . 4 |- ( x = z -> ( ( x e. A /\ ph ) <-> ( z e. A /\ [ z / x ] ph ) ) )
4		nfv 1508	. . . 4 |- F/ z ( x e. A /\ ph )
5		nfv 1508	. . . . 5 |- F/ x z e. A
6		nfs1v 1919	. . . . 5 |- F/ x [ z / x ] ph
7	5, 6	nfan 1545	. . . 4 |- F/ x ( z e. A /\ [ z / x ] ph )
8	3, 4, 7	cbviota 5139	. . 3 |- ( iota x ( x e. A /\ ph ) ) = ( iota z ( z e. A /\ [ z / x ] ph ) )
9		eleq1 2220	. . . . 5 |- ( z = y -> ( z e. A <-> y e. A ) )
10		sbequ 1820	. . . . . 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 1771	. . . . . 6 |- ( [ y / x ] ph <-> ps )
14	10, 13	bitrdi 195	. . . . 5 |- ( z = y -> ( [ z / x ] ph <-> ps ) )
15	9, 14	anbi12d 465	. . . 4 |- ( z = y -> ( ( z e. A /\ [ z / x ] ph ) <-> ( y e. A /\ ps ) ) )
16		nfv 1508	. . . . 5 |- F/ y z e. A
17		cbvriota.1	. . . . . 6 |- F/ y ph
18	17	nfsb 1926	. . . . 5 |- F/ y [ z / x ] ph
19	16, 18	nfan 1545	. . . 4 |- F/ y ( z e. A /\ [ z / x ] ph )
20		nfv 1508	. . . 4 |- F/ z ( y e. A /\ ps )
21	15, 19, 20	cbviota 5139	. . 3 |- ( iota z ( z e. A /\ [ z / x ] ph ) ) = ( iota y ( y e. A /\ ps ) )
22	8, 21	eqtri 2178	. 2 |- ( iota x ( x e. A /\ ph ) ) = ( iota y ( y e. A /\ ps ) )
23		df-riota 5777	. 2 |- ( iota_ x e. A ph ) = ( iota x ( x e. A /\ ph ) )
24		df-riota 5777	. 2 |- ( iota_ y e. A ps ) = ( iota y ( y e. A /\ ps ) )
25	22, 23, 24	3eqtr4i 2188	1 |- ( iota_ x e. A ph ) = ( iota_ y e. A ps )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1335   F/wnf 1440   [wsb 1742   e. wcel 2128   iotacio 5132   iota_crio 5776

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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rex 2441  df-sn 3566  df-uni 3773  df-iota 5134  df-riota 5777

This theorem is referenced by:  cbvriotav  5788