Theorem sbcnel12g 3110

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 2472	. . . 4 |- ( B e/ C <-> -. B e. C )
2	1	sbcbii 3058	. . 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 3039	. 2 |- ( A e. V -> ( [. A / x ]. -. B e. C <-> -. [. A / x ]. B e. C ) )
5		sbcel12g 3108	. . . 4 |- ( A e. V -> ( [. A / x ]. B e. C <-> [_ A / x ]_ B e. [_ A / x ]_ C ) )
6	5	notbid 669	. . 3 |- ( A e. V -> ( -. [. A / x ]. B e. C <-> -. [_ A / x ]_ B e. [_ A / x ]_ C ) )
7		df-nel 2472	. . 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 2176   e/ wnel 2471   [.wsbc 2998   [_csb 3093

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-nel 2472  df-v 2774  df-sbc 2999  df-csb 3094

This theorem is referenced by: (None)