Theorem cbvopab2v 4067

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 3767	. . . . . . 7 |- ( y = z -> <. x , y >. = <. x , z >. )
2	1	eqeq2d 2183	. . . . . 6 |- ( y = z -> ( w = <. x , y >. <-> w = <. x , z >. ) )
3		cbvopab2v.1	. . . . . 6 |- ( y = z -> ( ph <-> ps ) )
4	2, 3	anbi12d 471	. . . . 5 |- ( y = z -> ( ( w = <. x , y >. /\ ph ) <-> ( w = <. x , z >. /\ ps ) ) )
5	4	cbvexv 1912	. . . 4 |- ( E. y ( w = <. x , y >. /\ ph ) <-> E. z ( w = <. x , z >. /\ ps ) )
6	5	exbii 1599	. . 3 |- ( E. x E. y ( w = <. x , y >. /\ ph ) <-> E. x E. z ( w = <. x , z >. /\ ps ) )
7	6	abbii 2287	. 2 |- { w | E. x E. y ( w = <. x , y >. /\ ph ) } = { w | E. x E. z ( w = <. x , z >. /\ ps ) }
8		df-opab 4052	. 2 |- { <. x , y >. | ph } = { w | E. x E. y ( w = <. x , y >. /\ ph ) }
9		df-opab 4052	. 2 |- { <. x , z >. | ps } = { w | E. x E. z ( w = <. x , z >. /\ ps ) }
10	7, 8, 9	3eqtr4i 2202	1 |- { <. x , y >. | ph } = { <. x , z >. | ps }

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1349   E.wex 1486   {cab 2157   <.cop 3587   {copab 4050

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 705  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-10 1499  ax-11 1500  ax-i12 1501  ax-bndl 1503  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-ext 2153

This theorem depends on definitions:  df-bi 116  df-3an 976  df-tru 1352  df-nf 1455  df-sb 1757  df-clab 2158  df-cleq 2164  df-clel 2167  df-nfc 2302  df-v 2733  df-un 3126  df-sn 3590  df-pr 3591  df-op 3593  df-opab 4052

This theorem is referenced by: (None)