Theorem cbvopab2v 4727

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 4403	. . . . . . 7 |- ( y = z -> <. x , y >. = <. x , z >. )
2	1	eqeq2d 2632	. . . . . 6 |- ( y = z -> ( w = <. x , y >. <-> w = <. x , z >. ) )
3		cbvopab2v.1	. . . . . 6 |- ( y = z -> ( ph <-> ps ) )
4	2, 3	anbi12d 747	. . . . 5 |- ( y = z -> ( ( w = <. x , y >. /\ ph ) <-> ( w = <. x , z >. /\ ps ) ) )
5	4	cbvexv 2275	. . . 4 |- ( E. y ( w = <. x , y >. /\ ph ) <-> E. z ( w = <. x , z >. /\ ps ) )
6	5	exbii 1774	. . 3 |- ( E. x E. y ( w = <. x , y >. /\ ph ) <-> E. x E. z ( w = <. x , z >. /\ ps ) )
7	6	abbii 2739	. 2 |- { w | E. x E. y ( w = <. x , y >. /\ ph ) } = { w | E. x E. z ( w = <. x , z >. /\ ps ) }
8		df-opab 4713	. 2 |- { <. x , y >. | ph } = { w | E. x E. y ( w = <. x , y >. /\ ph ) }
9		df-opab 4713	. 2 |- { <. x , z >. | ps } = { w | E. x E. z ( w = <. x , z >. /\ ps ) }
10	7, 8, 9	3eqtr4i 2654	1 |- { <. x , y >. | ph } = { <. x , z >. | ps }

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   {cab 2608   <.cop 4183   {copab 4712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-opab 4713

This theorem is referenced by: (None)