Theorem sbcne12g 3102

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

Assertion

Ref	Expression
sbcne12g	|- ( A e. V -> ( [. A / x ]. B =/= C <-> [_ A / x ]_ B =/= [_ A / x ]_ C ) )

Proof of Theorem sbcne12g

Step	Hyp	Ref	Expression
1		sbceqg 3100	. . 3 |- ( A e. V -> ( [. A / x ]. B = C <-> [_ A / x ]_ B = [_ A / x ]_ C ) )
2	1	notbid 668	. 2 |- ( A e. V -> ( -. [. A / x ]. B = C <-> -. [_ A / x ]_ B = [_ A / x ]_ C ) )
3		df-ne 2368	. . . . 5 |- ( B =/= C <-> -. B = C )
4	3	sbcbii 3049	. . . 4 |- ( [. A / x ]. B =/= C <-> [. A / x ]. -. B = C )
5		sbcng 3030	. . . 4 |- ( A e. V -> ( [. A / x ]. -. B = C <-> -. [. A / x ]. B = C ) )
6	4, 5	bitrid 192	. . 3 |- ( A e. V -> ( [. A / x ]. B =/= C <-> -. [. A / x ]. B = C ) )
7		df-ne 2368	. . . 4 |- ( [_ A / x ]_ B =/= [_ A / x ]_ C <-> -. [_ A / x ]_ B = [_ A / x ]_ C )
8	7	a1i 9	. . 3 |- ( A e. V -> ( [_ A / x ]_ B =/= [_ A / x ]_ C <-> -. [_ A / x ]_ B = [_ A / x ]_ C ) )
9	6, 8	bibi12d 235	. 2 |- ( A e. V -> ( ( [. A / x ]. B =/= C <-> [_ A / x ]_ B =/= [_ A / x ]_ C ) <-> ( -. [. A / x ]. B = C <-> -. [_ A / x ]_ B = [_ A / x ]_ C ) ) )
10	2, 9	mpbird 167	1 |- ( A e. V -> ( [. A / x ]. B =/= C <-> [_ A / x ]_ B =/= [_ A / x ]_ C ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105   = wceq 1364   e. wcel 2167   =/= wne 2367   [.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-ne 2368  df-v 2765  df-sbc 2990  df-csb 3085

This theorem is referenced by: (None)