Theorem cbvoprab1 5969

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 1539	. . . . . 6 |- F/ w v = <. x , y >.
2		cbvoprab1.1	. . . . . 6 |- F/ w ph
3	1, 2	nfan 1576	. . . . 5 |- F/ w ( v = <. x , y >. /\ ph )
4	3	nfex 1648	. . . 4 |- F/ w E. y ( v = <. x , y >. /\ ph )
5		nfv 1539	. . . . . 6 |- F/ x v = <. w , y >.
6		cbvoprab1.2	. . . . . 6 |- F/ x ps
7	5, 6	nfan 1576	. . . . 5 |- F/ x ( v = <. w , y >. /\ ps )
8	7	nfex 1648	. . . 4 |- F/ x E. y ( v = <. w , y >. /\ ps )
9		opeq1 3793	. . . . . . 7 |- ( x = w -> <. x , y >. = <. w , y >. )
10	9	eqeq2d 2201	. . . . . 6 |- ( x = w -> ( v = <. x , y >. <-> v = <. w , y >. ) )
11		cbvoprab1.3	. . . . . 6 |- ( x = w -> ( ph <-> ps ) )
12	10, 11	anbi12d 473	. . . . 5 |- ( x = w -> ( ( v = <. x , y >. /\ ph ) <-> ( v = <. w , y >. /\ ps ) ) )
13	12	exbidv 1836	. . . 4 |- ( x = w -> ( E. y ( v = <. x , y >. /\ ph ) <-> E. y ( v = <. w , y >. /\ ps ) ) )
14	4, 8, 13	cbvex 1767	. . 3 |- ( E. x E. y ( v = <. x , y >. /\ ph ) <-> E. w E. y ( v = <. w , y >. /\ ps ) )
15	14	opabbii 4085	. 2 |- { <. v , z >. | E. x E. y ( v = <. x , y >. /\ ph ) } = { <. v , z >. | E. w E. y ( v = <. w , y >. /\ ps ) }
16		dfoprab2 5944	. 2 |- { <. <. x , y >. , z >. | ph } = { <. v , z >. | E. x E. y ( v = <. x , y >. /\ ph ) }
17		dfoprab2 5944	. 2 |- { <. <. w , y >. , z >. | ps } = { <. v , z >. | E. w E. y ( v = <. w , y >. /\ ps ) }
18	15, 16, 17	3eqtr4i 2220	1 |- { <. <. x , y >. , z >. | ph } = { <. <. w , y >. , z >. | ps }

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   F/wnf 1471   E.wex 1503   <.cop 3610   {copab 4078   {coprab 5898

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-opab 4080  df-oprab 5901

This theorem is referenced by: (None)