Theorem cbvopab1s

Description: Change first bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 31-Jul-2003)

	Ref	Expression
Assertion	cbvopab1s	|- { <. x , y >. | ph } = { <. z , y >. | [ z / x ] ph }

Proof

Step	Hyp	Ref	Expression
1		nfv	|- F/ z E. y ( w = <. x , y >. /\ ph )
2		nfv	|- F/ x w = <. z , y >.
3		nfs1v	|- F/ x [ z / x ] ph
4	2 3	nfan	|- F/ x ( w = <. z , y >. /\ [ z / x ] ph )
5	4	nfex	|- F/ x E. y ( w = <. z , y >. /\ [ z / x ] ph )
6		opeq1	|- ( x = z -> <. x , y >. = <. z , y >. )
7	6	eqeq2d	|- ( x = z -> ( w = <. x , y >. <-> w = <. z , y >. ) )
8		sbequ12	|- ( x = z -> ( ph <-> [ z / x ] ph ) )
9	7 8	anbi12d	|- ( x = z -> ( ( w = <. x , y >. /\ ph ) <-> ( w = <. z , y >. /\ [ z / x ] ph ) ) )
10	9	exbidv	|- ( x = z -> ( E. y ( w = <. x , y >. /\ ph ) <-> E. y ( w = <. z , y >. /\ [ z / x ] ph ) ) )
11	1 5 10	cbvexv1	|- ( E. x E. y ( w = <. x , y >. /\ ph ) <-> E. z E. y ( w = <. z , y >. /\ [ z / x ] ph ) )
12	11	abbii	|- { w | E. x E. y ( w = <. x , y >. /\ ph ) } = { w | E. z E. y ( w = <. z , y >. /\ [ z / x ] ph ) }
13		df-opab	|- { <. x , y >. | ph } = { w | E. x E. y ( w = <. x , y >. /\ ph ) }
14		df-opab	|- { <. z , y >. | [ z / x ] ph } = { w | E. z E. y ( w = <. z , y >. /\ [ z / x ] ph ) }
15	12 13 14	3eqtr4i	|- { <. x , y >. | ph } = { <. z , y >. | [ z / x ] ph }