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Metamath Proof Explorer

Theorem sbcnel12g

Description: Distribute proper substitution through negated membership. (Contributed by Andrew Salmon, 18-Jun-2011)

Ref Expression
Assertion sbcnel12g 
|- ( A e. V -> ( [. A / x ]. B e/ C <-> [_ A / x ]_ B e/ [_ A / x ]_ C ) )

Proof

Step Hyp Ref Expression
1 sbcng 
 |-  ( A e. V -> ( [. A / x ]. -. B e. C <-> -. [. A / x ]. B e. C ) )
2 df-nel 
 |-  ( B e/ C <-> -. B e. C )
3 2 sbcbii 
 |-  ( [. A / x ]. B e/ C <-> [. A / x ]. -. B e. C )
4 df-nel 
 |-  ( [_ A / x ]_ B e/ [_ A / x ]_ C <-> -. [_ A / x ]_ B e. [_ A / x ]_ C )
5 sbcel12 
 |-  ( [. A / x ]. B e. C <-> [_ A / x ]_ B e. [_ A / x ]_ C )
6 4 5 xchbinxr 
 |-  ( [_ A / x ]_ B e/ [_ A / x ]_ C <-> -. [. A / x ]. B e. C )
7 1 3 6 3bitr4g 
 |-  ( A e. V -> ( [. A / x ]. B e/ C <-> [_ A / x ]_ B e/ [_ A / x ]_ C ) )