Theorem cbvopab 3915

Description: Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 14-Sep-2003.)

Hypotheses

Ref	Expression
cbvopab.1	|- F/ z ph
cbvopab.2	|- F/ w ph
cbvopab.3	|- F/ x ps
cbvopab.4	|- F/ y ps
cbvopab.5	|- ( ( x = z /\ y = w ) -> ( ph <-> ps ) )

Assertion

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

Distinct variable group:   x, y, z, w

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

Proof of Theorem cbvopab

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1467	. . . . 5 |- F/ z v = <. x , y >.
2		cbvopab.1	. . . . 5 |- F/ z ph
3	1, 2	nfan 1503	. . . 4 |- F/ z ( v = <. x , y >. /\ ph )
4		nfv 1467	. . . . 5 |- F/ w v = <. x , y >.
5		cbvopab.2	. . . . 5 |- F/ w ph
6	4, 5	nfan 1503	. . . 4 |- F/ w ( v = <. x , y >. /\ ph )
7		nfv 1467	. . . . 5 |- F/ x v = <. z , w >.
8		cbvopab.3	. . . . 5 |- F/ x ps
9	7, 8	nfan 1503	. . . 4 |- F/ x ( v = <. z , w >. /\ ps )
10		nfv 1467	. . . . 5 |- F/ y v = <. z , w >.
11		cbvopab.4	. . . . 5 |- F/ y ps
12	10, 11	nfan 1503	. . . 4 |- F/ y ( v = <. z , w >. /\ ps )
13		opeq12 3630	. . . . . 6 |- ( ( x = z /\ y = w ) -> <. x , y >. = <. z , w >. )
14	13	eqeq2d 2100	. . . . 5 |- ( ( x = z /\ y = w ) -> ( v = <. x , y >. <-> v = <. z , w >. ) )
15		cbvopab.5	. . . . 5 |- ( ( x = z /\ y = w ) -> ( ph <-> ps ) )
16	14, 15	anbi12d 458	. . . 4 |- ( ( x = z /\ y = w ) -> ( ( v = <. x , y >. /\ ph ) <-> ( v = <. z , w >. /\ ps ) ) )
17	3, 6, 9, 12, 16	cbvex2 1846	. . 3 |- ( E. x E. y ( v = <. x , y >. /\ ph ) <-> E. z E. w ( v = <. z , w >. /\ ps ) )
18	17	abbii 2204	. 2 |- { v | E. x E. y ( v = <. x , y >. /\ ph ) } = { v | E. z E. w ( v = <. z , w >. /\ ps ) }
19		df-opab 3906	. 2 |- { <. x , y >. | ph } = { v | E. x E. y ( v = <. x , y >. /\ ph ) }
20		df-opab 3906	. 2 |- { <. z , w >. | ps } = { v | E. z E. w ( v = <. z , w >. /\ ps ) }
21	18, 19, 20	3eqtr4i 2119	1 |- { <. x , y >. | ph } = { <. z , w >. | ps }

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1290   F/wnf 1395   E.wex 1427   {cab 2075   <.cop 3453   {copab 3904

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-un 3004  df-sn 3456  df-pr 3457  df-op 3459  df-opab 3906

This theorem is referenced by:  cbvopabv  3916  opelopabsb  4096