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Metamath Proof Explorer

Theorem sbcralt

Description: Interchange class substitution and restricted quantifier. (Contributed by NM, 1-Mar-2008) (Revised by David Abernethy, 22-Feb-2010)

Ref Expression
Assertion sbcralt 
|- ( ( A e. V /\ F/_ y A ) -> ( [. A / x ]. A. y e. B ph <-> A. y e. B [. A / x ]. ph ) )

Proof

Step Hyp Ref Expression
1 sbccow 
 |-  ( [. A / z ]. [. z / x ]. A. y e. B ph <-> [. A / x ]. A. y e. B ph )
2 simpl 
 |-  ( ( A e. V /\ F/_ y A ) -> A e. V )
3 sbsbc 
 |-  ( [ z / x ] A. y e. B ph <-> [. z / x ]. A. y e. B ph )
4 nfcv 
 |-  F/_ x B
5 nfs1v 
 |-  F/ x [ z / x ] ph
6 4 5 nfralw 
 |-  F/ x A. y e. B [ z / x ] ph
7 sbequ12 
 |-  ( x = z -> ( ph <-> [ z / x ] ph ) )
8 7 ralbidv 
 |-  ( x = z -> ( A. y e. B ph <-> A. y e. B [ z / x ] ph ) )
9 6 8 sbiev 
 |-  ( [ z / x ] A. y e. B ph <-> A. y e. B [ z / x ] ph )
10 3 9 bitr3i 
 |-  ( [. z / x ]. A. y e. B ph <-> A. y e. B [ z / x ] ph )
11 nfnfc1 
 |-  F/ y F/_ y A
12 nfcvd 
 |-  ( F/_ y A -> F/_ y z )
13 id 
 |-  ( F/_ y A -> F/_ y A )
14 12 13 nfeqd 
 |-  ( F/_ y A -> F/ y z = A )
15 11 14 nfan1 
 |-  F/ y ( F/_ y A /\ z = A )
16 dfsbcq2 
 |-  ( z = A -> ( [ z / x ] ph <-> [. A / x ]. ph ) )
17 16 adantl 
 |-  ( ( F/_ y A /\ z = A ) -> ( [ z / x ] ph <-> [. A / x ]. ph ) )
18 15 17 ralbid 
 |-  ( ( F/_ y A /\ z = A ) -> ( A. y e. B [ z / x ] ph <-> A. y e. B [. A / x ]. ph ) )
19 18 adantll 
 |-  ( ( ( A e. V /\ F/_ y A ) /\ z = A ) -> ( A. y e. B [ z / x ] ph <-> A. y e. B [. A / x ]. ph ) )
20 10 19 bitrid 
 |-  ( ( ( A e. V /\ F/_ y A ) /\ z = A ) -> ( [. z / x ]. A. y e. B ph <-> A. y e. B [. A / x ]. ph ) )
21 2 20 sbcied 
 |-  ( ( A e. V /\ F/_ y A ) -> ( [. A / z ]. [. z / x ]. A. y e. B ph <-> A. y e. B [. A / x ]. ph ) )
22 1 21 bitr3id 
 |-  ( ( A e. V /\ F/_ y A ) -> ( [. A / x ]. A. y e. B ph <-> A. y e. B [. A / x ]. ph ) )