Theorem cbvoprab3 5929

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 1521	. . . . . 6 |- F/ w v = <. x , y >.
2		cbvoprab3.1	. . . . . 6 |- F/ w ph
3	1, 2	nfan 1558	. . . . 5 |- F/ w ( v = <. x , y >. /\ ph )
4	3	nfex 1630	. . . 4 |- F/ w E. y ( v = <. x , y >. /\ ph )
5	4	nfex 1630	. . 3 |- F/ w E. x E. y ( v = <. x , y >. /\ ph )
6		nfv 1521	. . . . . 6 |- F/ z v = <. x , y >.
7		cbvoprab3.2	. . . . . 6 |- F/ z ps
8	6, 7	nfan 1558	. . . . 5 |- F/ z ( v = <. x , y >. /\ ps )
9	8	nfex 1630	. . . 4 |- F/ z E. y ( v = <. x , y >. /\ ps )
10	9	nfex 1630	. . 3 |- F/ z E. x E. y ( v = <. x , y >. /\ ps )
11		cbvoprab3.3	. . . . 5 |- ( z = w -> ( ph <-> ps ) )
12	11	anbi2d 461	. . . 4 |- ( z = w -> ( ( v = <. x , y >. /\ ph ) <-> ( v = <. x , y >. /\ ps ) ) )
13	12	2exbidv 1861	. . 3 |- ( z = w -> ( E. x E. y ( v = <. x , y >. /\ ph ) <-> E. x E. y ( v = <. x , y >. /\ ps ) ) )
14	5, 10, 13	cbvopab2 4063	. 2 |- { <. v , z >. | E. x E. y ( v = <. x , y >. /\ ph ) } = { <. v , w >. | E. x E. y ( v = <. x , y >. /\ ps ) }
15		dfoprab2 5900	. 2 |- { <. <. x , y >. , z >. | ph } = { <. v , z >. | E. x E. y ( v = <. x , y >. /\ ph ) }
16		dfoprab2 5900	. 2 |- { <. <. x , y >. , w >. | ps } = { <. v , w >. | E. x E. y ( v = <. x , y >. /\ ps ) }
17	14, 15, 16	3eqtr4i 2201	1 |- { <. <. x , y >. , z >. | ph } = { <. <. x , y >. , w >. | ps }

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   F/wnf 1453   E.wex 1485   <.cop 3586   {copab 4049   {coprab 5854

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-opab 4051  df-oprab 5857

This theorem is referenced by:  cbvoprab3v  5930  tposoprab  6259  erovlem  6605