Theorem sbcbi1 3000

Description: Distribution of class substitution over biconditional. One direction of sbcbig 2997 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)

Assertion

Ref	Expression
sbcbi1	|- ( [. A / x ]. ( ph <-> ps ) -> ( [. A / x ]. ph <-> [. A / x ]. ps ) )

Proof of Theorem sbcbi1

Step	Hyp	Ref	Expression
1		sbcex 2959	. 2 |- ( [. A / x ]. ( ph <-> ps ) -> A e. _V )
2		sbcbig 2997	. . 3 |- ( A e. _V -> ( [. A / x ]. ( ph <-> ps ) <-> ( [. A / x ]. ph <-> [. A / x ]. ps ) ) )
3	2	biimpd 143	. 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   <-> wb 104   e. wcel 2136   _Vcvv 2726   [.wsbc 2951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-sbc 2952

This theorem is referenced by: (None)