Theorem cbvopab2 4490

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 1772	. . . . . 6 |- F/ z w = <. x , y >.
2		cbvopab2.1	. . . . . 6 |- F/ z ph
3	1, 2	nfan 2022	. . . . 5 |- F/ z ( w = <. x , y >. /\ ph )
4		nfv 1772	. . . . . 6 |- F/ y w = <. x , z >.
5		cbvopab2.2	. . . . . 6 |- F/ y ps
6	4, 5	nfan 2022	. . . . 5 |- F/ y ( w = <. x , z >. /\ ps )
7		opeq2 4181	. . . . . . 7 |- ( y = z -> <. x , y >. = <. x , z >. )
8	7	eqeq2d 2472	. . . . . 6 |- ( y = z -> ( w = <. x , y >. <-> w = <. x , z >. ) )
9		cbvopab2.3	. . . . . 6 |- ( y = z -> ( ph <-> ps ) )
10	8, 9	anbi12d 722	. . . . 5 |- ( y = z -> ( ( w = <. x , y >. /\ ph ) <-> ( w = <. x , z >. /\ ps ) ) )
11	3, 6, 10	cbvex 2126	. . . 4 |- ( E. y ( w = <. x , y >. /\ ph ) <-> E. z ( w = <. x , z >. /\ ps ) )
12	11	exbii 1729	. . 3 |- ( E. x E. y ( w = <. x , y >. /\ ph ) <-> E. x E. z ( w = <. x , z >. /\ ps ) )
13	12	abbii 2578	. 2 |- { w | E. x E. y ( w = <. x , y >. /\ ph ) } = { w | E. x E. z ( w = <. x , z >. /\ ps ) }
14		df-opab 4478	. 2 |- { <. x , y >. | ph } = { w | E. x E. y ( w = <. x , y >. /\ ph ) }
15		df-opab 4478	. 2 |- { <. x , z >. | ps } = { w | E. x E. z ( w = <. x , z >. /\ ps ) }
16	13, 14, 15	3eqtr4i 2494	1 |- { <. x , y >. | ph } = { <. x , z >. | ps }

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 375   = wceq 1455   E.wex 1674   F/wnf 1678   {cab 2448   <.cop 3986   {copab 4476

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-rab 2758  df-v 3059  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-opab 4478

This theorem is referenced by:  cbvoprab3  6399