Theorem cbvopab1 4009

Description: Change first bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 6-Oct-2004.) (Revised by Mario Carneiro, 14-Oct-2016.)

Hypotheses

Ref	Expression
cbvopab1.1	|- F/ z ph
cbvopab1.2	|- F/ x ps
cbvopab1.3	|- ( x = z -> ( ph <-> ps ) )

Assertion

Ref	Expression
cbvopab1	|- { <. x , y >. | ph } = { <. z , y >. | ps }

Distinct variable groups:   x, y   y, z

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

Proof of Theorem cbvopab1

Dummy variables w v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1509	. . . . 5 |- F/ v E. y ( w = <. x , y >. /\ ph )
2		nfv 1509	. . . . . . 7 |- F/ x w = <. v , y >.
3		nfs1v 1913	. . . . . . 7 |- F/ x [ v / x ] ph
4	2, 3	nfan 1545	. . . . . 6 |- F/ x ( w = <. v , y >. /\ [ v / x ] ph )
5	4	nfex 1617	. . . . 5 |- F/ x E. y ( w = <. v , y >. /\ [ v / x ] ph )
6		opeq1 3713	. . . . . . . 8 |- ( x = v -> <. x , y >. = <. v , y >. )
7	6	eqeq2d 2152	. . . . . . 7 |- ( x = v -> ( w = <. x , y >. <-> w = <. v , y >. ) )
8		sbequ12 1745	. . . . . . 7 |- ( x = v -> ( ph <-> [ v / x ] ph ) )
9	7, 8	anbi12d 465	. . . . . 6 |- ( x = v -> ( ( w = <. x , y >. /\ ph ) <-> ( w = <. v , y >. /\ [ v / x ] ph ) ) )
10	9	exbidv 1798	. . . . 5 |- ( x = v -> ( E. y ( w = <. x , y >. /\ ph ) <-> E. y ( w = <. v , y >. /\ [ v / x ] ph ) ) )
11	1, 5, 10	cbvex 1730	. . . 4 |- ( E. x E. y ( w = <. x , y >. /\ ph ) <-> E. v E. y ( w = <. v , y >. /\ [ v / x ] ph ) )
12		nfv 1509	. . . . . . 7 |- F/ z w = <. v , y >.
13		cbvopab1.1	. . . . . . . 8 |- F/ z ph
14	13	nfsb 1920	. . . . . . 7 |- F/ z [ v / x ] ph
15	12, 14	nfan 1545	. . . . . 6 |- F/ z ( w = <. v , y >. /\ [ v / x ] ph )
16	15	nfex 1617	. . . . 5 |- F/ z E. y ( w = <. v , y >. /\ [ v / x ] ph )
17		nfv 1509	. . . . 5 |- F/ v E. y ( w = <. z , y >. /\ ps )
18		opeq1 3713	. . . . . . . 8 |- ( v = z -> <. v , y >. = <. z , y >. )
19	18	eqeq2d 2152	. . . . . . 7 |- ( v = z -> ( w = <. v , y >. <-> w = <. z , y >. ) )
20		sbequ 1813	. . . . . . . 8 |- ( v = z -> ( [ v / x ] ph <-> [ z / x ] ph ) )
21		cbvopab1.2	. . . . . . . . 9 |- F/ x ps
22		cbvopab1.3	. . . . . . . . 9 |- ( x = z -> ( ph <-> ps ) )
23	21, 22	sbie 1765	. . . . . . . 8 |- ( [ z / x ] ph <-> ps )
24	20, 23	syl6bb 195	. . . . . . 7 |- ( v = z -> ( [ v / x ] ph <-> ps ) )
25	19, 24	anbi12d 465	. . . . . 6 |- ( v = z -> ( ( w = <. v , y >. /\ [ v / x ] ph ) <-> ( w = <. z , y >. /\ ps ) ) )
26	25	exbidv 1798	. . . . 5 |- ( v = z -> ( E. y ( w = <. v , y >. /\ [ v / x ] ph ) <-> E. y ( w = <. z , y >. /\ ps ) ) )
27	16, 17, 26	cbvex 1730	. . . 4 |- ( E. v E. y ( w = <. v , y >. /\ [ v / x ] ph ) <-> E. z E. y ( w = <. z , y >. /\ ps ) )
28	11, 27	bitri 183	. . 3 |- ( E. x E. y ( w = <. x , y >. /\ ph ) <-> E. z E. y ( w = <. z , y >. /\ ps ) )
29	28	abbii 2256	. 2 |- { w | E. x E. y ( w = <. x , y >. /\ ph ) } = { w | E. z E. y ( w = <. z , y >. /\ ps ) }
30		df-opab 3998	. 2 |- { <. x , y >. | ph } = { w | E. x E. y ( w = <. x , y >. /\ ph ) }
31		df-opab 3998	. 2 |- { <. z , y >. | ps } = { w | E. z E. y ( w = <. z , y >. /\ ps ) }
32	29, 30, 31	3eqtr4i 2171	1 |- { <. x , y >. | ph } = { <. z , y >. | ps }

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1332   F/wnf 1437   E.wex 1469   [wsb 1736   {cab 2126   <.cop 3535   {copab 3996

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3080  df-sn 3538  df-pr 3539  df-op 3541  df-opab 3998

This theorem is referenced by:  cbvopab1v  4012  cbvmptf  4030  cbvmpt  4031