Theorem cbvoprab3 5918

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 1516	. . . . . 6 |- F/ w v = <. x , y >.
2		cbvoprab3.1	. . . . . 6 |- F/ w ph
3	1, 2	nfan 1553	. . . . 5 |- F/ w ( v = <. x , y >. /\ ph )
4	3	nfex 1625	. . . 4 |- F/ w E. y ( v = <. x , y >. /\ ph )
5	4	nfex 1625	. . 3 |- F/ w E. x E. y ( v = <. x , y >. /\ ph )
6		nfv 1516	. . . . . 6 |- F/ z v = <. x , y >.
7		cbvoprab3.2	. . . . . 6 |- F/ z ps
8	6, 7	nfan 1553	. . . . 5 |- F/ z ( v = <. x , y >. /\ ps )
9	8	nfex 1625	. . . 4 |- F/ z E. y ( v = <. x , y >. /\ ps )
10	9	nfex 1625	. . 3 |- F/ z E. x E. y ( v = <. x , y >. /\ ps )
11		cbvoprab3.3	. . . . 5 |- ( z = w -> ( ph <-> ps ) )
12	11	anbi2d 460	. . . 4 |- ( z = w -> ( ( v = <. x , y >. /\ ph ) <-> ( v = <. x , y >. /\ ps ) ) )
13	12	2exbidv 1856	. . 3 |- ( z = w -> ( E. x E. y ( v = <. x , y >. /\ ph ) <-> E. x E. y ( v = <. x , y >. /\ ps ) ) )
14	5, 10, 13	cbvopab2 4056	. 2 |- { <. v , z >. | E. x E. y ( v = <. x , y >. /\ ph ) } = { <. v , w >. | E. x E. y ( v = <. x , y >. /\ ps ) }
15		dfoprab2 5889	. 2 |- { <. <. x , y >. , z >. | ph } = { <. v , z >. | E. x E. y ( v = <. x , y >. /\ ph ) }
16		dfoprab2 5889	. 2 |- { <. <. x , y >. , w >. | ps } = { <. v , w >. | E. x E. y ( v = <. x , y >. /\ ps ) }
17	14, 15, 16	3eqtr4i 2196	1 |- { <. <. x , y >. , z >. | ph } = { <. <. x , y >. , w >. | ps }

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   F/wnf 1448   E.wex 1480   <.cop 3579   {copab 4042   {coprab 5843

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-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-opab 4044  df-oprab 5846

This theorem is referenced by:  cbvoprab3v  5919  tposoprab  6248  erovlem  6593