Theorem sbcnel12g 3101

Description: Distribute proper substitution through negated membership. (Contributed by Andrew Salmon, 18-Jun-2011.)

Assertion

Ref	Expression
sbcnel12g	|- ( A e. V -> ( [. A / x ]. B e/ C <-> [_ A / x ]_ B e/ [_ A / x ]_ C ) )

Proof of Theorem sbcnel12g

Step	Hyp	Ref	Expression
1		df-nel 2463	. . . 4 |- ( B e/ C <-> -. B e. C )
2	1	sbcbii 3049	. . 3 |- ( [. A / x ]. B e/ C <-> [. A / x ]. -. B e. C )
3	2	a1i 9	. 2 |- ( A e. V -> ( [. A / x ]. B e/ C <-> [. A / x ]. -. B e. C ) )
4		sbcng 3030	. 2 |- ( A e. V -> ( [. A / x ]. -. B e. C <-> -. [. A / x ]. B e. C ) )
5		sbcel12g 3099	. . . 4 |- ( A e. V -> ( [. A / x ]. B e. C <-> [_ A / x ]_ B e. [_ A / x ]_ C ) )
6	5	notbid 668	. . 3 |- ( A e. V -> ( -. [. A / x ]. B e. C <-> -. [_ A / x ]_ B e. [_ A / x ]_ C ) )
7		df-nel 2463	. . 3 |- ( [_ A / x ]_ B e/ [_ A / x ]_ C <-> -. [_ A / x ]_ B e. [_ A / x ]_ C )
8	6, 7	bitr4di 198	. 2 |- ( A e. V -> ( -. [. A / x ]. B e. C <-> [_ A / x ]_ B e/ [_ A / x ]_ C ) )
9	3, 4, 8	3bitrd 214	1 |- ( A e. V -> ( [. A / x ]. B e/ C <-> [_ A / x ]_ B e/ [_ A / x ]_ C ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105   e. wcel 2167   e/ wnel 2462   [.wsbc 2989   [_csb 3084

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-nel 2463  df-v 2765  df-sbc 2990  df-csb 3085

This theorem is referenced by: (None)