Theorem cbvopab1s 4159

Description: Change first bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 31-Jul-2003.)

Assertion

Ref	Expression
cbvopab1s	|- { <. x , y >. | ph } = { <. z , y >. | [ z / x ] ph }

Distinct variable groups:   x, y, z   ph, z

Allowed substitution hints:   ph( x, y)

Proof of Theorem cbvopab1s

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1574	. . . 4 |- F/ z E. y ( w = <. x , y >. /\ ph )
2		nfv 1574	. . . . . 6 |- F/ x w = <. z , y >.
3		nfs1v 1990	. . . . . 6 |- F/ x [ z / x ] ph
4	2, 3	nfan 1611	. . . . 5 |- F/ x ( w = <. z , y >. /\ [ z / x ] ph )
5	4	nfex 1683	. . . 4 |- F/ x E. y ( w = <. z , y >. /\ [ z / x ] ph )
6		opeq1 3857	. . . . . . 7 |- ( x = z -> <. x , y >. = <. z , y >. )
7	6	eqeq2d 2241	. . . . . 6 |- ( x = z -> ( w = <. x , y >. <-> w = <. z , y >. ) )
8		sbequ12 1817	. . . . . 6 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
9	7, 8	anbi12d 473	. . . . 5 |- ( x = z -> ( ( w = <. x , y >. /\ ph ) <-> ( w = <. z , y >. /\ [ z / x ] ph ) ) )
10	9	exbidv 1871	. . . 4 |- ( x = z -> ( E. y ( w = <. x , y >. /\ ph ) <-> E. y ( w = <. z , y >. /\ [ z / x ] ph ) ) )
11	1, 5, 10	cbvex 1802	. . 3 |- ( E. x E. y ( w = <. x , y >. /\ ph ) <-> E. z E. y ( w = <. z , y >. /\ [ z / x ] ph ) )
12	11	abbii 2345	. 2 |- { w | E. x E. y ( w = <. x , y >. /\ ph ) } = { w | E. z E. y ( w = <. z , y >. /\ [ z / x ] ph ) }
13		df-opab 4146	. 2 |- { <. x , y >. | ph } = { w | E. x E. y ( w = <. x , y >. /\ ph ) }
14		df-opab 4146	. 2 |- { <. z , y >. | [ z / x ] ph } = { w | E. z E. y ( w = <. z , y >. /\ [ z / x ] ph ) }
15	12, 13, 14	3eqtr4i 2260	1 |- { <. x , y >. | ph } = { <. z , y >. | [ z / x ] ph }

Syntax hints:   /\ wa 104   = wceq 1395   E.wex 1538   [wsb 1808   {cab 2215   <.cop 3669   {copab 4144

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675  df-opab 4146

This theorem is referenced by: (None)