Theorem cbvopab 3828

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 1421	. . . . 5 |- F/ z v = <. x , y >.
2		cbvopab.1	. . . . 5 |- F/ z ph
3	1, 2	nfan 1457	. . . 4 |- F/ z ( v = <. x , y >. /\ ph )
4		nfv 1421	. . . . 5 |- F/ w v = <. x , y >.
5		cbvopab.2	. . . . 5 |- F/ w ph
6	4, 5	nfan 1457	. . . 4 |- F/ w ( v = <. x , y >. /\ ph )
7		nfv 1421	. . . . 5 |- F/ x v = <. z , w >.
8		cbvopab.3	. . . . 5 |- F/ x ps
9	7, 8	nfan 1457	. . . 4 |- F/ x ( v = <. z , w >. /\ ps )
10		nfv 1421	. . . . 5 |- F/ y v = <. z , w >.
11		cbvopab.4	. . . . 5 |- F/ y ps
12	10, 11	nfan 1457	. . . 4 |- F/ y ( v = <. z , w >. /\ ps )
13		opeq12 3551	. . . . . 6 |- ( ( x = z /\ y = w ) -> <. x , y >. = <. z , w >. )
14	13	eqeq2d 2051	. . . . 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 442	. . . 4 |- ( ( x = z /\ y = w ) -> ( ( v = <. x , y >. /\ ph ) <-> ( v = <. z , w >. /\ ps ) ) )
17	3, 6, 9, 12, 16	cbvex2 1797	. . 3 |- ( E. x E. y ( v = <. x , y >. /\ ph ) <-> E. z E. w ( v = <. z , w >. /\ ps ) )
18	17	abbii 2153	. 2 |- { v | E. x E. y ( v = <. x , y >. /\ ph ) } = { v | E. z E. w ( v = <. z , w >. /\ ps ) }
19		df-opab 3819	. 2 |- { <. x , y >. | ph } = { v | E. x E. y ( v = <. x , y >. /\ ph ) }
20		df-opab 3819	. 2 |- { <. z , w >. | ps } = { v | E. z E. w ( v = <. z , w >. /\ ps ) }
21	18, 19, 20	3eqtr4i 2070	1 |- { <. x , y >. | ph } = { <. z , w >. | ps }

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   F/wnf 1349   E.wex 1381   {cab 2026   <.cop 3378   {copab 3817

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-opab 3819

This theorem is referenced by:  cbvopabv  3829  opelopabsb  3997