Theorem cbvoprab2 5597

Description: Change the second bound variable in an operation abstraction. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 11-Dec-2016.)

Hypotheses

Ref	Expression
cbvoprab2.1	|- F/ w ph
cbvoprab2.2	|- F/ y ps
cbvoprab2.3	|- ( y = w -> ( ph <-> ps ) )

Assertion

Ref	Expression
cbvoprab2	|- { <. <. x , y >. , z >. | ph } = { <. <. x , w >. , z >. | ps }

Distinct variable group:   x, w, y, z

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

Proof of Theorem cbvoprab2

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1461	. . . . . . 7 |- F/ w v = <. <. x , y >. , z >.
2		cbvoprab2.1	. . . . . . 7 |- F/ w ph
3	1, 2	nfan 1497	. . . . . 6 |- F/ w ( v = <. <. x , y >. , z >. /\ ph )
4	3	nfex 1568	. . . . 5 |- F/ w E. z ( v = <. <. x , y >. , z >. /\ ph )
5		nfv 1461	. . . . . . 7 |- F/ y v = <. <. x , w >. , z >.
6		cbvoprab2.2	. . . . . . 7 |- F/ y ps
7	5, 6	nfan 1497	. . . . . 6 |- F/ y ( v = <. <. x , w >. , z >. /\ ps )
8	7	nfex 1568	. . . . 5 |- F/ y E. z ( v = <. <. x , w >. , z >. /\ ps )
9		opeq2 3571	. . . . . . . . 9 |- ( y = w -> <. x , y >. = <. x , w >. )
10	9	opeq1d 3576	. . . . . . . 8 |- ( y = w -> <. <. x , y >. , z >. = <. <. x , w >. , z >. )
11	10	eqeq2d 2092	. . . . . . 7 |- ( y = w -> ( v = <. <. x , y >. , z >. <-> v = <. <. x , w >. , z >. ) )
12		cbvoprab2.3	. . . . . . 7 |- ( y = w -> ( ph <-> ps ) )
13	11, 12	anbi12d 456	. . . . . 6 |- ( y = w -> ( ( v = <. <. x , y >. , z >. /\ ph ) <-> ( v = <. <. x , w >. , z >. /\ ps ) ) )
14	13	exbidv 1746	. . . . 5 |- ( y = w -> ( E. z ( v = <. <. x , y >. , z >. /\ ph ) <-> E. z ( v = <. <. x , w >. , z >. /\ ps ) ) )
15	4, 8, 14	cbvex 1679	. . . 4 |- ( E. y E. z ( v = <. <. x , y >. , z >. /\ ph ) <-> E. w E. z ( v = <. <. x , w >. , z >. /\ ps ) )
16	15	exbii 1536	. . 3 |- ( E. x E. y E. z ( v = <. <. x , y >. , z >. /\ ph ) <-> E. x E. w E. z ( v = <. <. x , w >. , z >. /\ ps ) )
17	16	abbii 2194	. 2 |- { v | E. x E. y E. z ( v = <. <. x , y >. , z >. /\ ph ) } = { v | E. x E. w E. z ( v = <. <. x , w >. , z >. /\ ps ) }
18		df-oprab 5536	. 2 |- { <. <. x , y >. , z >. | ph } = { v | E. x E. y E. z ( v = <. <. x , y >. , z >. /\ ph ) }
19		df-oprab 5536	. 2 |- { <. <. x , w >. , z >. | ps } = { v | E. x E. w E. z ( v = <. <. x , w >. , z >. /\ ps ) }
20	17, 18, 19	3eqtr4i 2111	1 |- { <. <. x , y >. , z >. | ph } = { <. <. x , w >. , z >. | ps }

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   F/wnf 1389   E.wex 1421   {cab 2067   <.cop 3401   {coprab 5533

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-sn 3404  df-pr 3405  df-op 3407  df-oprab 5536

This theorem is referenced by: (None)