Theorem cbvopab1s 4080

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 1528	. . . 4 |- F/ z E. y ( w = <. x , y >. /\ ph )
2		nfv 1528	. . . . . 6 |- F/ x w = <. z , y >.
3		nfs1v 1939	. . . . . 6 |- F/ x [ z / x ] ph
4	2, 3	nfan 1565	. . . . 5 |- F/ x ( w = <. z , y >. /\ [ z / x ] ph )
5	4	nfex 1637	. . . 4 |- F/ x E. y ( w = <. z , y >. /\ [ z / x ] ph )
6		opeq1 3780	. . . . . . 7 |- ( x = z -> <. x , y >. = <. z , y >. )
7	6	eqeq2d 2189	. . . . . 6 |- ( x = z -> ( w = <. x , y >. <-> w = <. z , y >. ) )
8		sbequ12 1771	. . . . . 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 1825	. . . 4 |- ( x = z -> ( E. y ( w = <. x , y >. /\ ph ) <-> E. y ( w = <. z , y >. /\ [ z / x ] ph ) ) )
11	1, 5, 10	cbvex 1756	. . 3 |- ( E. x E. y ( w = <. x , y >. /\ ph ) <-> E. z E. y ( w = <. z , y >. /\ [ z / x ] ph ) )
12	11	abbii 2293	. 2 |- { w | E. x E. y ( w = <. x , y >. /\ ph ) } = { w | E. z E. y ( w = <. z , y >. /\ [ z / x ] ph ) }
13		df-opab 4067	. 2 |- { <. x , y >. | ph } = { w | E. x E. y ( w = <. x , y >. /\ ph ) }
14		df-opab 4067	. 2 |- { <. z , y >. | [ z / x ] ph } = { w | E. z E. y ( w = <. z , y >. /\ [ z / x ] ph ) }
15	12, 13, 14	3eqtr4i 2208	1 |- { <. x , y >. | ph } = { <. z , y >. | [ z / x ] ph }

Syntax hints:   /\ wa 104   = wceq 1353   E.wex 1492   [wsb 1762   {cab 2163   <.cop 3597   {copab 4065

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-un 3135  df-sn 3600  df-pr 3601  df-op 3603  df-opab 4067

This theorem is referenced by: (None)