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Metamath Proof Explorer

Theorem sbcimg

Description: Distribution of class substitution over implication. (Contributed by NM, 16-Jan-2004)

Ref Expression
Assertion sbcimg 
|- ( A e. V -> ( [. A / x ]. ( ph -> ps ) <-> ( [. A / x ]. ph -> [. A / x ]. ps ) ) )

Proof

Step Hyp Ref Expression
1 dfsbcq2 
 |-  ( y = A -> ( [ y / x ] ( ph -> ps ) <-> [. A / x ]. ( ph -> ps ) ) )
2 dfsbcq2 
 |-  ( y = A -> ( [ y / x ] ph <-> [. A / x ]. ph ) )
3 dfsbcq2 
 |-  ( y = A -> ( [ y / x ] ps <-> [. A / x ]. ps ) )
4 2 3 imbi12d 
 |-  ( y = A -> ( ( [ y / x ] ph -> [ y / x ] ps ) <-> ( [. A / x ]. ph -> [. A / x ]. ps ) ) )
5 sbim 
 |-  ( [ y / x ] ( ph -> ps ) <-> ( [ y / x ] ph -> [ y / x ] ps ) )
6 1 4 5 vtoclbg 
 |-  ( A e. V -> ( [. A / x ]. ( ph -> ps ) <-> ( [. A / x ]. ph -> [. A / x ]. ps ) ) )