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Metamath Proof Explorer

Theorem sbcim1

Description: Distribution of class substitution over implication. One direction of sbcimg that holds for proper classes. (Contributed by NM, 17-Aug-2018) Avoid ax-10 , ax-12 . (Revised by SN, 26-Oct-2024)

Ref Expression
Assertion sbcim1 
|- ( [. A / x ]. ( ph -> ps ) -> ( [. A / x ]. ph -> [. A / x ]. ps ) )

Proof

Step Hyp Ref Expression
1 sbcex 
 |-  ( [. A / x ]. ( ph -> ps ) -> A e. _V )
2 dfsbcq2 
 |-  ( y = A -> ( [ y / x ] ( ph -> ps ) <-> [. A / x ]. ( ph -> ps ) ) )
3 dfsbcq2 
 |-  ( y = A -> ( [ y / x ] ph <-> [. A / x ]. ph ) )
4 dfsbcq2 
 |-  ( y = A -> ( [ y / x ] ps <-> [. A / x ]. ps ) )
5 3 4 imbi12d 
 |-  ( y = A -> ( ( [ y / x ] ph -> [ y / x ] ps ) <-> ( [. A / x ]. ph -> [. A / x ]. ps ) ) )
6 2 5 imbi12d 
 |-  ( y = A -> ( ( [ y / x ] ( ph -> ps ) -> ( [ y / x ] ph -> [ y / x ] ps ) ) <-> ( [. A / x ]. ( ph -> ps ) -> ( [. A / x ]. ph -> [. A / x ]. ps ) ) ) )
7 sbi1 
 |-  ( [ y / x ] ( ph -> ps ) -> ( [ y / x ] ph -> [ y / x ] ps ) )
8 6 7 vtoclg 
 |-  ( A e. _V -> ( [. A / x ]. ( ph -> ps ) -> ( [. A / x ]. ph -> [. A / x ]. ps ) ) )
9 1 8 mpcom 
 |-  ( [. A / x ]. ( ph -> ps ) -> ( [. A / x ]. ph -> [. A / x ]. ps ) )