Theorem sbcng 2944

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 2907	. 2 |- ( y = A -> ( [ y / x ] -. ph <-> [. A / x ]. -. ph ) )
2		dfsbcq2 2907	. . 3 |- ( y = A -> ( [ y / x ] ph <-> [. A / x ]. ph ) )
3	2	notbid 656	. 2 |- ( y = A -> ( -. [ y / x ] ph <-> -. [. A / x ]. ph ) )
4		sbn 1923	. 2 |- ( [ y / x ] -. ph <-> -. [ y / x ] ph )
5	1, 3, 4	vtoclbg 2742	1 |- ( A e. V -> ( [. A / x ]. -. ph <-> -. [. A / x ]. ph ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 104   = wceq 1331   e. wcel 1480   [wsb 1735   [.wsbc 2904

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-sbc 2905

This theorem is referenced by:  sbcn1  2951  sbcnel12g  3014  sbcne12g  3015  difopab  4667  zsupcllemstep  11627