Theorem cbvoprab3 5580

Description: Rule used to change the third bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 22-Aug-2013.)

Hypotheses

Ref	Expression
cbvoprab3.1	|- F/ w ph
cbvoprab3.2	|- F/ z ps
cbvoprab3.3	|- ( z = w -> ( ph <-> ps ) )

Assertion

Ref	Expression
cbvoprab3	|- { <. <. x , y >. , z >. | ph } = { <. <. x , y >. , w >. | ps }

Distinct variable groups:   x, z, w   y, z, w

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

Proof of Theorem cbvoprab3

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1421	. . . . . 6 |- F/ w v = <. x , y >.
2		cbvoprab3.1	. . . . . 6 |- F/ w ph
3	1, 2	nfan 1457	. . . . 5 |- F/ w ( v = <. x , y >. /\ ph )
4	3	nfex 1528	. . . 4 |- F/ w E. y ( v = <. x , y >. /\ ph )
5	4	nfex 1528	. . 3 |- F/ w E. x E. y ( v = <. x , y >. /\ ph )
6		nfv 1421	. . . . . 6 |- F/ z v = <. x , y >.
7		cbvoprab3.2	. . . . . 6 |- F/ z ps
8	6, 7	nfan 1457	. . . . 5 |- F/ z ( v = <. x , y >. /\ ps )
9	8	nfex 1528	. . . 4 |- F/ z E. y ( v = <. x , y >. /\ ps )
10	9	nfex 1528	. . 3 |- F/ z E. x E. y ( v = <. x , y >. /\ ps )
11		cbvoprab3.3	. . . . 5 |- ( z = w -> ( ph <-> ps ) )
12	11	anbi2d 437	. . . 4 |- ( z = w -> ( ( v = <. x , y >. /\ ph ) <-> ( v = <. x , y >. /\ ps ) ) )
13	12	2exbidv 1748	. . 3 |- ( z = w -> ( E. x E. y ( v = <. x , y >. /\ ph ) <-> E. x E. y ( v = <. x , y >. /\ ps ) ) )
14	5, 10, 13	cbvopab2 3831	. 2 |- { <. v , z >. | E. x E. y ( v = <. x , y >. /\ ph ) } = { <. v , w >. | E. x E. y ( v = <. x , y >. /\ ps ) }
15		dfoprab2 5552	. 2 |- { <. <. x , y >. , z >. | ph } = { <. v , z >. | E. x E. y ( v = <. x , y >. /\ ph ) }
16		dfoprab2 5552	. 2 |- { <. <. x , y >. , w >. | ps } = { <. v , w >. | E. x E. y ( v = <. x , y >. /\ ps ) }
17	14, 15, 16	3eqtr4i 2070	1 |- { <. <. x , y >. , z >. | ph } = { <. <. x , y >. , w >. | ps }

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   F/wnf 1349   E.wex 1381   <.cop 3378   {copab 3817   {coprab 5513

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-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

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-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-opab 3819  df-oprab 5516

This theorem is referenced by:  cbvoprab3v  5581  tposoprab  5895  erovlem  6198