Theorem cbvoprab3 6021

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 1551	. . . . . 6 |- F/ w v = <. x , y >.
2		cbvoprab3.1	. . . . . 6 |- F/ w ph
3	1, 2	nfan 1588	. . . . 5 |- F/ w ( v = <. x , y >. /\ ph )
4	3	nfex 1660	. . . 4 |- F/ w E. y ( v = <. x , y >. /\ ph )
5	4	nfex 1660	. . 3 |- F/ w E. x E. y ( v = <. x , y >. /\ ph )
6		nfv 1551	. . . . . 6 |- F/ z v = <. x , y >.
7		cbvoprab3.2	. . . . . 6 |- F/ z ps
8	6, 7	nfan 1588	. . . . 5 |- F/ z ( v = <. x , y >. /\ ps )
9	8	nfex 1660	. . . 4 |- F/ z E. y ( v = <. x , y >. /\ ps )
10	9	nfex 1660	. . 3 |- F/ z E. x E. y ( v = <. x , y >. /\ ps )
11		cbvoprab3.3	. . . . 5 |- ( z = w -> ( ph <-> ps ) )
12	11	anbi2d 464	. . . 4 |- ( z = w -> ( ( v = <. x , y >. /\ ph ) <-> ( v = <. x , y >. /\ ps ) ) )
13	12	2exbidv 1891	. . 3 |- ( z = w -> ( E. x E. y ( v = <. x , y >. /\ ph ) <-> E. x E. y ( v = <. x , y >. /\ ps ) ) )
14	5, 10, 13	cbvopab2 4118	. 2 |- { <. v , z >. | E. x E. y ( v = <. x , y >. /\ ph ) } = { <. v , w >. | E. x E. y ( v = <. x , y >. /\ ps ) }
15		dfoprab2 5992	. 2 |- { <. <. x , y >. , z >. | ph } = { <. v , z >. | E. x E. y ( v = <. x , y >. /\ ph ) }
16		dfoprab2 5992	. 2 |- { <. <. x , y >. , w >. | ps } = { <. v , w >. | E. x E. y ( v = <. x , y >. /\ ps ) }
17	14, 15, 16	3eqtr4i 2236	1 |- { <. <. x , y >. , z >. | ph } = { <. <. x , y >. , w >. | ps }

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   F/wnf 1483   E.wex 1515   <.cop 3636   {copab 4104   {coprab 5945

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-opab 4106  df-oprab 5948

This theorem is referenced by:  cbvoprab3v  6022  tposoprab  6366  erovlem  6714