Theorem sbcnel12g 3773

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		sbcng 3307	. 2 |- ( A e. V -> ( [. A / x ]. -. B e. C <-> -. [. A / x ]. B e. C ) )
2		df-nel 2624	. . 3 |- ( B e/ C <-> -. B e. C )
3	2	sbcbii 3322	. 2 |- ( [. A / x ]. B e/ C <-> [. A / x ]. -. B e. C )
4		df-nel 2624	. . 3 |- ( [_ A / x ]_ B e/ [_ A / x ]_ C <-> -. [_ A / x ]_ B e. [_ A / x ]_ C )
5		sbcel12 3771	. . 3 |- ( [. A / x ]. B e. C <-> [_ A / x ]_ B e. [_ A / x ]_ C )
6	4, 5	xchbinxr 313	. 2 |- ( [_ A / x ]_ B e/ [_ A / x ]_ C <-> -. [. A / x ]. B e. C )
7	1, 3, 6	3bitr4g 292	1 |- ( A e. V -> ( [. A / x ]. B e/ C <-> [_ A / x ]_ B e/ [_ A / x ]_ C ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   e. wcel 1886   e/ wnel 2622   [.wsbc 3266   [_csb 3362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1446  df-fal 1449  df-ex 1663  df-nf 1667  df-sb 1797  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-nel 2624  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-in 3410  df-ss 3417  df-nul 3731

This theorem is referenced by:  rusbcALT  36784