Theorem cbvopab1s 4369

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 1673	. . . 4 |- F/ z E. y ( w = <. x , y >. /\ ph )
2		nfv 1673	. . . . . 6 |- F/ x w = <. z , y >.
3		nfs1v 2142	. . . . . 6 |- F/ x [ z / x ] ph
4	2, 3	nfan 1861	. . . . 5 |- F/ x ( w = <. z , y >. /\ [ z / x ] ph )
5	4	nfex 1874	. . . 4 |- F/ x E. y ( w = <. z , y >. /\ [ z / x ] ph )
6		opeq1 4064	. . . . . . 7 |- ( x = z -> <. x , y >. = <. z , y >. )
7	6	eqeq2d 2454	. . . . . 6 |- ( x = z -> ( w = <. x , y >. <-> w = <. z , y >. ) )
8		sbequ12 1936	. . . . . 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 1680	. . . 4 |- ( x = z -> ( E. y ( w = <. x , y >. /\ ph ) <-> E. y ( w = <. z , y >. /\ [ z / x ] ph ) ) )
11	1, 5, 10	cbvex 1970	. . 3 |- ( E. x E. y ( w = <. x , y >. /\ ph ) <-> E. z E. y ( w = <. z , y >. /\ [ z / x ] ph ) )
12	11	abbii 2560	. 2 |- { w | E. x E. y ( w = <. x , y >. /\ ph ) } = { w | E. z E. y ( w = <. z , y >. /\ [ z / x ] ph ) }
13		df-opab 4356	. 2 |- { <. x , y >. | ph } = { w | E. x E. y ( w = <. x , y >. /\ ph ) }
14		df-opab 4356	. 2 |- { <. z , y >. | [ z / x ] ph } = { w | E. z E. y ( w = <. z , y >. /\ [ z / x ] ph ) }
15	12, 13, 14	3eqtr4i 2473	1 |- { <. x , y >. | ph } = { <. z , y >. | [ z / x ] ph }

Syntax hints:   /\ wa 369   = wceq 1369   E.wex 1586   [wsb 1700   {cab 2429   <.cop 3888   {copab 4354

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-rab 2729  df-v 2979  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-opab 4356

This theorem is referenced by: (None)