Theorem cbvopab2v 4166

Description: Rule used to change the second bound variable in an ordered pair abstraction, using implicit substitution. (Contributed by NM, 2-Sep-1999.)

Hypothesis

Ref	Expression
cbvopab2v.1	|- ( y = z -> ( ph <-> ps ) )

Assertion

Ref	Expression
cbvopab2v	|- { <. x , y >. | ph } = { <. x , z >. | ps }

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

Allowed substitution hints:   ph( x, y)   ps( x, z)

Proof of Theorem cbvopab2v

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opeq2 3863	. . . . . . 7 |- ( y = z -> <. x , y >. = <. x , z >. )
2	1	eqeq2d 2243	. . . . . 6 |- ( y = z -> ( w = <. x , y >. <-> w = <. x , z >. ) )
3		cbvopab2v.1	. . . . . 6 |- ( y = z -> ( ph <-> ps ) )
4	2, 3	anbi12d 473	. . . . 5 |- ( y = z -> ( ( w = <. x , y >. /\ ph ) <-> ( w = <. x , z >. /\ ps ) ) )
5	4	cbvexv 1967	. . . 4 |- ( E. y ( w = <. x , y >. /\ ph ) <-> E. z ( w = <. x , z >. /\ ps ) )
6	5	exbii 1653	. . 3 |- ( E. x E. y ( w = <. x , y >. /\ ph ) <-> E. x E. z ( w = <. x , z >. /\ ps ) )
7	6	abbii 2347	. 2 |- { w | E. x E. y ( w = <. x , y >. /\ ph ) } = { w | E. x E. z ( w = <. x , z >. /\ ps ) }
8		df-opab 4151	. 2 |- { <. x , y >. | ph } = { w | E. x E. y ( w = <. x , y >. /\ ph ) }
9		df-opab 4151	. 2 |- { <. x , z >. | ps } = { w | E. x E. z ( w = <. x , z >. /\ ps ) }
10	7, 8, 9	3eqtr4i 2262	1 |- { <. x , y >. | ph } = { <. x , z >. | ps }

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   E.wex 1540   {cab 2217   <.cop 3672   {copab 4149

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676  df-op 3678  df-opab 4151

This theorem is referenced by: (None)