Theorem cbvoprab3 6731

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 1843	. . . . . 6 |- F/ w v = <. x , y >.
2		cbvoprab3.1	. . . . . 6 |- F/ w ph
3	1, 2	nfan 1828	. . . . 5 |- F/ w ( v = <. x , y >. /\ ph )
4	3	nfex 2154	. . . 4 |- F/ w E. y ( v = <. x , y >. /\ ph )
5	4	nfex 2154	. . 3 |- F/ w E. x E. y ( v = <. x , y >. /\ ph )
6		nfv 1843	. . . . . 6 |- F/ z v = <. x , y >.
7		cbvoprab3.2	. . . . . 6 |- F/ z ps
8	6, 7	nfan 1828	. . . . 5 |- F/ z ( v = <. x , y >. /\ ps )
9	8	nfex 2154	. . . 4 |- F/ z E. y ( v = <. x , y >. /\ ps )
10	9	nfex 2154	. . 3 |- F/ z E. x E. y ( v = <. x , y >. /\ ps )
11		cbvoprab3.3	. . . . 5 |- ( z = w -> ( ph <-> ps ) )
12	11	anbi2d 740	. . . 4 |- ( z = w -> ( ( v = <. x , y >. /\ ph ) <-> ( v = <. x , y >. /\ ps ) ) )
13	12	2exbidv 1852	. . 3 |- ( z = w -> ( E. x E. y ( v = <. x , y >. /\ ph ) <-> E. x E. y ( v = <. x , y >. /\ ps ) ) )
14	5, 10, 13	cbvopab2 4724	. 2 |- { <. v , z >. | E. x E. y ( v = <. x , y >. /\ ph ) } = { <. v , w >. | E. x E. y ( v = <. x , y >. /\ ps ) }
15		dfoprab2 6701	. 2 |- { <. <. x , y >. , z >. | ph } = { <. v , z >. | E. x E. y ( v = <. x , y >. /\ ph ) }
16		dfoprab2 6701	. 2 |- { <. <. x , y >. , w >. | ps } = { <. v , w >. | E. x E. y ( v = <. x , y >. /\ ps ) }
17	14, 15, 16	3eqtr4i 2654	1 |- { <. <. x , y >. , z >. | ph } = { <. <. x , y >. , w >. | ps }

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   F/wnf 1708   <.cop 4183   {copab 4712   {coprab 6651

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-opab 4713  df-oprab 6654

This theorem is referenced by:  cbvoprab3v  6732  tposoprab  7388  erovlem  7843