Theorem sbcn1 3313

Description: Move negation in and out of class substitution. One direction of sbcng 3308 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 3277	. 2 |- ( [. A / x ]. -. ph -> A e. _V )
2		sbcng 3308	. . 3 |- ( A e. _V -> ( [. A / x ]. -. ph <-> -. [. A / x ]. ph ) )
3	2	biimpd 211	. 2 |- ( A e. _V -> ( [. A / x ]. -. ph -> -. [. A / x ]. ph ) )
4	1, 3	mpcom 37	1 |- ( [. A / x ]. -. ph -> -. [. A / x ]. ph )

Syntax hints:   -. wn 3   -> wi 4   e. wcel 1887   _Vcvv 3045   [.wsbc 3267

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-12 1933  ax-13 2091  ax-ext 2431

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-v 3047  df-sbc 3268

This theorem is referenced by: (None)