Theorem cbvopab1s 4475

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 1761	. . . 4 |- F/ z E. y ( w = <. x , y >. /\ ph )
2		nfv 1761	. . . . . 6 |- F/ x w = <. z , y >.
3		nfs1v 2266	. . . . . 6 |- F/ x [ z / x ] ph
4	2, 3	nfan 2011	. . . . 5 |- F/ x ( w = <. z , y >. /\ [ z / x ] ph )
5	4	nfex 2031	. . . 4 |- F/ x E. y ( w = <. z , y >. /\ [ z / x ] ph )
6		opeq1 4166	. . . . . . 7 |- ( x = z -> <. x , y >. = <. z , y >. )
7	6	eqeq2d 2461	. . . . . 6 |- ( x = z -> ( w = <. x , y >. <-> w = <. z , y >. ) )
8		sbequ12 2083	. . . . . 6 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
9	7, 8	anbi12d 717	. . . . 5 |- ( x = z -> ( ( w = <. x , y >. /\ ph ) <-> ( w = <. z , y >. /\ [ z / x ] ph ) ) )
10	9	exbidv 1768	. . . 4 |- ( x = z -> ( E. y ( w = <. x , y >. /\ ph ) <-> E. y ( w = <. z , y >. /\ [ z / x ] ph ) ) )
11	1, 5, 10	cbvex 2115	. . 3 |- ( E. x E. y ( w = <. x , y >. /\ ph ) <-> E. z E. y ( w = <. z , y >. /\ [ z / x ] ph ) )
12	11	abbii 2567	. 2 |- { w | E. x E. y ( w = <. x , y >. /\ ph ) } = { w | E. z E. y ( w = <. z , y >. /\ [ z / x ] ph ) }
13		df-opab 4462	. 2 |- { <. x , y >. | ph } = { w | E. x E. y ( w = <. x , y >. /\ ph ) }
14		df-opab 4462	. 2 |- { <. z , y >. | [ z / x ] ph } = { w | E. z E. y ( w = <. z , y >. /\ [ z / x ] ph ) }
15	12, 13, 14	3eqtr4i 2483	1 |- { <. x , y >. | ph } = { <. z , y >. | [ z / x ] ph }

Syntax hints:   /\ wa 371   = wceq 1444   E.wex 1663   [wsb 1797   {cab 2437   <.cop 3974   {copab 4460

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-rab 2746  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-opab 4462

This theorem is referenced by: (None)