Theorem sbcim1 3047

Description: Distribution of class substitution over implication. One direction of sbcimg 3040 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)

Assertion

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

Proof of Theorem sbcim1

Step	Hyp	Ref	Expression
1		sbcex 3007	. 2 |- ( [. A / x ]. ( ph -> ps ) -> A e. _V )
2		sbcimg 3040	. . 3 |- ( A e. _V -> ( [. A / x ]. ( ph -> ps ) <-> ( [. A / x ]. ph -> [. A / x ]. ps ) ) )
3	2	biimpd 144	. 2 |- ( A e. _V -> ( [. A / x ]. ( ph -> ps ) -> ( [. A / x ]. ph -> [. A / x ]. ps ) ) )
4	1, 3	mpcom 36	1 |- ( [. A / x ]. ( ph -> ps ) -> ( [. A / x ]. ph -> [. A / x ]. ps ) )

Syntax hints:   -> wi 4   e. wcel 2176   _Vcvv 2772   [.wsbc 2998

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-sbc 2999

This theorem is referenced by:  sbcimdv  3064