Theorem sbcn1 3012

Description: Move negation in and out of class substitution. One direction of sbcng 3005 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 2973	. 2 |- ( [. A / x ]. -. ph -> A e. _V )
2		sbcng 3005	. . 3 |- ( A e. _V -> ( [. A / x ]. -. ph <-> -. [. A / x ]. ph ) )
3	2	biimpd 144	. 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 2148   _Vcvv 2739   [.wsbc 2964

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-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-sbc 2965

This theorem is referenced by: (None)