Theorem cbvopab1s 4529

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 1708	. . . 4 |- F/ z E. y ( w = <. x , y >. /\ ph )
2		nfv 1708	. . . . . 6 |- F/ x w = <. z , y >.
3		nfs1v 2182	. . . . . 6 |- F/ x [ z / x ] ph
4	2, 3	nfan 1929	. . . . 5 |- F/ x ( w = <. z , y >. /\ [ z / x ] ph )
5	4	nfex 1949	. . . 4 |- F/ x E. y ( w = <. z , y >. /\ [ z / x ] ph )
6		opeq1 4219	. . . . . . 7 |- ( x = z -> <. x , y >. = <. z , y >. )
7	6	eqeq2d 2471	. . . . . 6 |- ( x = z -> ( w = <. x , y >. <-> w = <. z , y >. ) )
8		sbequ12 1993	. . . . . 6 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
9	7, 8	anbi12d 710	. . . . 5 |- ( x = z -> ( ( w = <. x , y >. /\ ph ) <-> ( w = <. z , y >. /\ [ z / x ] ph ) ) )
10	9	exbidv 1715	. . . 4 |- ( x = z -> ( E. y ( w = <. x , y >. /\ ph ) <-> E. y ( w = <. z , y >. /\ [ z / x ] ph ) ) )
11	1, 5, 10	cbvex 2023	. . 3 |- ( E. x E. y ( w = <. x , y >. /\ ph ) <-> E. z E. y ( w = <. z , y >. /\ [ z / x ] ph ) )
12	11	abbii 2591	. 2 |- { w | E. x E. y ( w = <. x , y >. /\ ph ) } = { w | E. z E. y ( w = <. z , y >. /\ [ z / x ] ph ) }
13		df-opab 4516	. 2 |- { <. x , y >. | ph } = { w | E. x E. y ( w = <. x , y >. /\ ph ) }
14		df-opab 4516	. 2 |- { <. z , y >. | [ z / x ] ph } = { w | E. z E. y ( w = <. z , y >. /\ [ z / x ] ph ) }
15	12, 13, 14	3eqtr4i 2496	1 |- { <. x , y >. | ph } = { <. z , y >. | [ z / x ] ph }

Syntax hints:   /\ wa 369   = wceq 1395   E.wex 1613   [wsb 1740   {cab 2442   <.cop 4038   {copab 4514

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-opab 4516

This theorem is referenced by: (None)