Theorem sbcn1 2908

Description: Move negation in and out of class substitution. One direction of sbcng 2901 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)

Assertion

Ref	Expression
sbcn1	|- ( [. A / x ]. -. ph -> -. [. A / x ]. ph )

Proof of Theorem sbcn1

Step	Hyp	Ref	Expression
1		sbcex 2870	. 2 |- ( [. A / x ]. -. ph -> A e. _V )
2		sbcng 2901	. . 3 |- ( A e. _V -> ( [. A / x ]. -. ph <-> -. [. A / x ]. ph ) )
3	2	biimpd 143	. 2 |- ( A e. _V -> ( [. A / x ]. -. ph -> -. [. A / x ]. ph ) )
4	1, 3	mpcom 36	1 |- ( [. A / x ]. -. ph -> -. [. A / x ]. ph )

Syntax hints:   -. wn 3   -> wi 4   e. wcel 1448   _Vcvv 2641   [.wsbc 2862

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-sbc 2863

This theorem is referenced by: (None)