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Metamath Proof Explorer

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 }