Theorem cbvopab2v 4043

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 3744	. . . . . . 7 |- ( y = z -> <. x , y >. = <. x , z >. )
2	1	eqeq2d 2169	. . . . . 6 |- ( y = z -> ( w = <. x , y >. <-> w = <. x , z >. ) )
3		cbvopab2v.1	. . . . . 6 |- ( y = z -> ( ph <-> ps ) )
4	2, 3	anbi12d 465	. . . . 5 |- ( y = z -> ( ( w = <. x , y >. /\ ph ) <-> ( w = <. x , z >. /\ ps ) ) )
5	4	cbvexv 1898	. . . 4 |- ( E. y ( w = <. x , y >. /\ ph ) <-> E. z ( w = <. x , z >. /\ ps ) )
6	5	exbii 1585	. . 3 |- ( E. x E. y ( w = <. x , y >. /\ ph ) <-> E. x E. z ( w = <. x , z >. /\ ps ) )
7	6	abbii 2273	. 2 |- { w | E. x E. y ( w = <. x , y >. /\ ph ) } = { w | E. x E. z ( w = <. x , z >. /\ ps ) }
8		df-opab 4028	. 2 |- { <. x , y >. | ph } = { w | E. x E. y ( w = <. x , y >. /\ ph ) }
9		df-opab 4028	. 2 |- { <. x , z >. | ps } = { w | E. x E. z ( w = <. x , z >. /\ ps ) }
10	7, 8, 9	3eqtr4i 2188	1 |- { <. x , y >. | ph } = { <. x , z >. | ps }

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1335   E.wex 1472   {cab 2143   <.cop 3564   {copab 4026

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-sn 3567  df-pr 3568  df-op 3570  df-opab 4028

This theorem is referenced by: (None)