Theorem cbvrald 13669

Description: Rule used to change bound variables, using implicit substitution. (Contributed by BJ, 22-Nov-2019.)

Hypotheses

Ref	Expression
cbvrald.nf0	|- F/ x ph
cbvrald.nf1	|- F/ y ph
cbvrald.nf2	|- ( ph -> F/ y ps )
cbvrald.nf3	|- ( ph -> F/ x ch )
cbvrald.is	|- ( ph -> ( x = y -> ( ps <-> ch ) ) )

Assertion

Ref	Expression
cbvrald	|- ( ph -> ( A. x e. A ps <-> A. y e. A ch ) )

Distinct variable groups:   x, A   y, A

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

Proof of Theorem cbvrald

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbvrald.nf0	. . . 4 |- F/ x ph
2		nfv 1516	. . . 4 |- F/ z ph
3		nfv 1516	. . . . . 6 |- F/ z x e. A
4	3	a1i 9	. . . . 5 |- ( ph -> F/ z x e. A )
5		nfv 1516	. . . . . 6 |- F/ z ps
6	5	a1i 9	. . . . 5 |- ( ph -> F/ z ps )
7	4, 6	nfimd 1573	. . . 4 |- ( ph -> F/ z ( x e. A -> ps ) )
8		nfv 1516	. . . . . 6 |- F/ x z e. A
9	8	a1i 9	. . . . 5 |- ( ph -> F/ x z e. A )
10		nfs1v 1927	. . . . . 6 |- F/ x [ z / x ] ps
11	10	a1i 9	. . . . 5 |- ( ph -> F/ x [ z / x ] ps )
12	9, 11	nfimd 1573	. . . 4 |- ( ph -> F/ x ( z e. A -> [ z / x ] ps ) )
13		eleq1 2229	. . . . . . 7 |- ( x = z -> ( x e. A <-> z e. A ) )
14	13	adantl 275	. . . . . 6 |- ( ( ph /\ x = z ) -> ( x e. A <-> z e. A ) )
15		sbequ12 1759	. . . . . . 7 |- ( x = z -> ( ps <-> [ z / x ] ps ) )
16	15	adantl 275	. . . . . 6 |- ( ( ph /\ x = z ) -> ( ps <-> [ z / x ] ps ) )
17	14, 16	imbi12d 233	. . . . 5 |- ( ( ph /\ x = z ) -> ( ( x e. A -> ps ) <-> ( z e. A -> [ z / x ] ps ) ) )
18	17	ex 114	. . . 4 |- ( ph -> ( x = z -> ( ( x e. A -> ps ) <-> ( z e. A -> [ z / x ] ps ) ) ) )
19	1, 2, 7, 12, 18	cbv2 1737	. . 3 |- ( ph -> ( A. x ( x e. A -> ps ) <-> A. z ( z e. A -> [ z / x ] ps ) ) )
20		cbvrald.nf1	. . . 4 |- F/ y ph
21		nfv 1516	. . . . . 6 |- F/ y z e. A
22	21	a1i 9	. . . . 5 |- ( ph -> F/ y z e. A )
23		cbvrald.nf2	. . . . . 6 |- ( ph -> F/ y ps )
24	1, 23	nfsbd 1965	. . . . 5 |- ( ph -> F/ y [ z / x ] ps )
25	22, 24	nfimd 1573	. . . 4 |- ( ph -> F/ y ( z e. A -> [ z / x ] ps ) )
26		nfv 1516	. . . . . 6 |- F/ z y e. A
27	26	a1i 9	. . . . 5 |- ( ph -> F/ z y e. A )
28		nfv 1516	. . . . . 6 |- F/ z ch
29	28	a1i 9	. . . . 5 |- ( ph -> F/ z ch )
30	27, 29	nfimd 1573	. . . 4 |- ( ph -> F/ z ( y e. A -> ch ) )
31		eleq1 2229	. . . . . . 7 |- ( z = y -> ( z e. A <-> y e. A ) )
32	31	adantl 275	. . . . . 6 |- ( ( ph /\ z = y ) -> ( z e. A <-> y e. A ) )
33		sbequ 1828	. . . . . . 7 |- ( z = y -> ( [ z / x ] ps <-> [ y / x ] ps ) )
34		cbvrald.nf3	. . . . . . . 8 |- ( ph -> F/ x ch )
35		cbvrald.is	. . . . . . . 8 |- ( ph -> ( x = y -> ( ps <-> ch ) ) )
36	1, 34, 35	sbied 1776	. . . . . . 7 |- ( ph -> ( [ y / x ] ps <-> ch ) )
37	33, 36	sylan9bbr 459	. . . . . 6 |- ( ( ph /\ z = y ) -> ( [ z / x ] ps <-> ch ) )
38	32, 37	imbi12d 233	. . . . 5 |- ( ( ph /\ z = y ) -> ( ( z e. A -> [ z / x ] ps ) <-> ( y e. A -> ch ) ) )
39	38	ex 114	. . . 4 |- ( ph -> ( z = y -> ( ( z e. A -> [ z / x ] ps ) <-> ( y e. A -> ch ) ) ) )
40	2, 20, 25, 30, 39	cbv2 1737	. . 3 |- ( ph -> ( A. z ( z e. A -> [ z / x ] ps ) <-> A. y ( y e. A -> ch ) ) )
41	19, 40	bitrd 187	. 2 |- ( ph -> ( A. x ( x e. A -> ps ) <-> A. y ( y e. A -> ch ) ) )
42		df-ral 2449	. 2 |- ( A. x e. A ps <-> A. x ( x e. A -> ps ) )
43		df-ral 2449	. 2 |- ( A. y e. A ch <-> A. y ( y e. A -> ch ) )
44	41, 42, 43	3bitr4g 222	1 |- ( ph -> ( A. x e. A ps <-> A. y e. A ch ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1341   F/wnf 1448   [wsb 1750   e. wcel 2136   A.wral 2444

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-cleq 2158  df-clel 2161  df-ral 2449

This theorem is referenced by:  setindft  13847