Theorem cbvoprab1 5596

Description: Rule used to change first bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 20-Dec-2008.) (Revised by Mario Carneiro, 5-Dec-2016.)

Hypotheses

Ref	Expression
cbvoprab1.1	|- F/ w ph
cbvoprab1.2	|- F/ x ps
cbvoprab1.3	|- ( x = w -> ( ph <-> ps ) )

Assertion

Ref	Expression
cbvoprab1	|- { <. <. x , y >. , z >. | ph } = { <. <. w , y >. , z >. | ps }

Distinct variable group:   x, y, z, w

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

Proof of Theorem cbvoprab1

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1461	. . . . . 6 |- F/ w v = <. x , y >.
2		cbvoprab1.1	. . . . . 6 |- F/ w ph
3	1, 2	nfan 1497	. . . . 5 |- F/ w ( v = <. x , y >. /\ ph )
4	3	nfex 1568	. . . 4 |- F/ w E. y ( v = <. x , y >. /\ ph )
5		nfv 1461	. . . . . 6 |- F/ x v = <. w , y >.
6		cbvoprab1.2	. . . . . 6 |- F/ x ps
7	5, 6	nfan 1497	. . . . 5 |- F/ x ( v = <. w , y >. /\ ps )
8	7	nfex 1568	. . . 4 |- F/ x E. y ( v = <. w , y >. /\ ps )
9		opeq1 3570	. . . . . . 7 |- ( x = w -> <. x , y >. = <. w , y >. )
10	9	eqeq2d 2092	. . . . . 6 |- ( x = w -> ( v = <. x , y >. <-> v = <. w , y >. ) )
11		cbvoprab1.3	. . . . . 6 |- ( x = w -> ( ph <-> ps ) )
12	10, 11	anbi12d 456	. . . . 5 |- ( x = w -> ( ( v = <. x , y >. /\ ph ) <-> ( v = <. w , y >. /\ ps ) ) )
13	12	exbidv 1746	. . . 4 |- ( x = w -> ( E. y ( v = <. x , y >. /\ ph ) <-> E. y ( v = <. w , y >. /\ ps ) ) )
14	4, 8, 13	cbvex 1679	. . 3 |- ( E. x E. y ( v = <. x , y >. /\ ph ) <-> E. w E. y ( v = <. w , y >. /\ ps ) )
15	14	opabbii 3845	. 2 |- { <. v , z >. | E. x E. y ( v = <. x , y >. /\ ph ) } = { <. v , z >. | E. w E. y ( v = <. w , y >. /\ ps ) }
16		dfoprab2 5572	. 2 |- { <. <. x , y >. , z >. | ph } = { <. v , z >. | E. x E. y ( v = <. x , y >. /\ ph ) }
17		dfoprab2 5572	. 2 |- { <. <. w , y >. , z >. | ps } = { <. v , z >. | E. w E. y ( v = <. w , y >. /\ ps ) }
18	15, 16, 17	3eqtr4i 2111	1 |- { <. <. x , y >. , z >. | ph } = { <. <. w , y >. , z >. | ps }

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   F/wnf 1389   E.wex 1421   <.cop 3401   {copab 3838   {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-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

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-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-opab 3840  df-oprab 5536

This theorem is referenced by: (None)