Theorem sbcng 3476

Description: Move negation in and out of class substitution. (Contributed by NM, 16-Jan-2004.)

Assertion

Ref	Expression
sbcng	|- ( A e. V -> ( [. A / x ]. -. ph <-> -. [. A / x ]. ph ) )

Proof of Theorem sbcng

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq2 3438	. 2 |- ( y = A -> ( [ y / x ] -. ph <-> [. A / x ]. -. ph ) )
2		dfsbcq2 3438	. . 3 |- ( y = A -> ( [ y / x ] ph <-> [. A / x ]. ph ) )
3	2	notbid 308	. 2 |- ( y = A -> ( -. [ y / x ] ph <-> -. [. A / x ]. ph ) )
4		sbn 2391	. 2 |- ( [ y / x ] -. ph <-> -. [ y / x ] ph )
5	1, 3, 4	vtoclbg 3267	1 |- ( A e. V -> ( [. A / x ]. -. ph <-> -. [. A / x ]. ph ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   = wceq 1483   [wsb 1880   e. wcel 1990   [.wsbc 3435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202  df-sbc 3436

This theorem is referenced by:  sbcn1  3481  sbcrext  3511  sbcrextOLD  3512  sbcnel12g  3985  sbcne12  3986  difopab  5253  bnj23  30784  bnj110  30928  bnj1204  31080  sbcni  33914  frege124d  38053  onfrALTlem5  38757  onfrALTlem5VD  39121