Theorem sbcng 3030

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 2992	. 2 |- ( y = A -> ( [ y / x ] -. ph <-> [. A / x ]. -. ph ) )
2		dfsbcq2 2992	. . 3 |- ( y = A -> ( [ y / x ] ph <-> [. A / x ]. ph ) )
3	2	notbid 668	. 2 |- ( y = A -> ( -. [ y / x ] ph <-> -. [. A / x ]. ph ) )
4		sbn 1971	. 2 |- ( [ y / x ] -. ph <-> -. [ y / x ] ph )
5	1, 3, 4	vtoclbg 2825	1 |- ( A e. V -> ( [. A / x ]. -. ph <-> -. [. A / x ]. ph ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105   = wceq 1364   [wsb 1776   e. wcel 2167   [.wsbc 2989

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-sbc 2990

This theorem is referenced by:  sbcn1  3037  sbcnel12g  3101  sbcne12g  3102  difopab  4799  zsupcllemstep  10319