Theorem cbvopab2 3872

Description: Change second bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 22-Aug-2013.)

Hypotheses

Ref	Expression
cbvopab2.1	|- F/ z ph
cbvopab2.2	|- F/ y ps
cbvopab2.3	|- ( y = z -> ( ph <-> ps ) )

Assertion

Ref	Expression
cbvopab2	|- { <. x , y >. | ph } = { <. x , z >. | ps }

Distinct variable group:   x, y, z

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

Proof of Theorem cbvopab2

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1462	. . . . . 6 |- F/ z w = <. x , y >.
2		cbvopab2.1	. . . . . 6 |- F/ z ph
3	1, 2	nfan 1498	. . . . 5 |- F/ z ( w = <. x , y >. /\ ph )
4		nfv 1462	. . . . . 6 |- F/ y w = <. x , z >.
5		cbvopab2.2	. . . . . 6 |- F/ y ps
6	4, 5	nfan 1498	. . . . 5 |- F/ y ( w = <. x , z >. /\ ps )
7		opeq2 3591	. . . . . . 7 |- ( y = z -> <. x , y >. = <. x , z >. )
8	7	eqeq2d 2094	. . . . . 6 |- ( y = z -> ( w = <. x , y >. <-> w = <. x , z >. ) )
9		cbvopab2.3	. . . . . 6 |- ( y = z -> ( ph <-> ps ) )
10	8, 9	anbi12d 457	. . . . 5 |- ( y = z -> ( ( w = <. x , y >. /\ ph ) <-> ( w = <. x , z >. /\ ps ) ) )
11	3, 6, 10	cbvex 1681	. . . 4 |- ( E. y ( w = <. x , y >. /\ ph ) <-> E. z ( w = <. x , z >. /\ ps ) )
12	11	exbii 1537	. . 3 |- ( E. x E. y ( w = <. x , y >. /\ ph ) <-> E. x E. z ( w = <. x , z >. /\ ps ) )
13	12	abbii 2198	. 2 |- { w | E. x E. y ( w = <. x , y >. /\ ph ) } = { w | E. x E. z ( w = <. x , z >. /\ ps ) }
14		df-opab 3860	. 2 |- { <. x , y >. | ph } = { w | E. x E. y ( w = <. x , y >. /\ ph ) }
15		df-opab 3860	. 2 |- { <. x , z >. | ps } = { w | E. x E. z ( w = <. x , z >. /\ ps ) }
16	13, 14, 15	3eqtr4i 2113	1 |- { <. x , y >. | ph } = { <. x , z >. | ps }

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1285   F/wnf 1390   E.wex 1422   {cab 2069   <.cop 3419   {copab 3858

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-un 2986  df-sn 3422  df-pr 3423  df-op 3425  df-opab 3860

This theorem is referenced by:  cbvoprab3  5632