Theorem cbvoprab1 6092

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 1576	. . . . . 6 |- F/ w v = <. x , y >.
2		cbvoprab1.1	. . . . . 6 |- F/ w ph
3	1, 2	nfan 1613	. . . . 5 |- F/ w ( v = <. x , y >. /\ ph )
4	3	nfex 1685	. . . 4 |- F/ w E. y ( v = <. x , y >. /\ ph )
5		nfv 1576	. . . . . 6 |- F/ x v = <. w , y >.
6		cbvoprab1.2	. . . . . 6 |- F/ x ps
7	5, 6	nfan 1613	. . . . 5 |- F/ x ( v = <. w , y >. /\ ps )
8	7	nfex 1685	. . . 4 |- F/ x E. y ( v = <. w , y >. /\ ps )
9		opeq1 3862	. . . . . . 7 |- ( x = w -> <. x , y >. = <. w , y >. )
10	9	eqeq2d 2243	. . . . . 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 1873	. . . 4 |- ( x = w -> ( E. y ( v = <. x , y >. /\ ph ) <-> E. y ( v = <. w , y >. /\ ps ) ) )
14	4, 8, 13	cbvex 1804	. . 3 |- ( E. x E. y ( v = <. x , y >. /\ ph ) <-> E. w E. y ( v = <. w , y >. /\ ps ) )
15	14	opabbii 4156	. 2 |- { <. v , z >. | E. x E. y ( v = <. x , y >. /\ ph ) } = { <. v , z >. | E. w E. y ( v = <. w , y >. /\ ps ) }
16		dfoprab2 6067	. 2 |- { <. <. x , y >. , z >. | ph } = { <. v , z >. | E. x E. y ( v = <. x , y >. /\ ph ) }
17		dfoprab2 6067	. 2 |- { <. <. w , y >. , z >. | ps } = { <. v , z >. | E. w E. y ( v = <. w , y >. /\ ps ) }
18	15, 16, 17	3eqtr4i 2262	1 |- { <. <. x , y >. , z >. | ph } = { <. <. w , y >. , z >. | ps }

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   F/wnf 1508   E.wex 1540   <.cop 3672   {copab 4149   {coprab 6018

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-opab 4151  df-oprab 6021

This theorem is referenced by: (None)