Theorem sbcne12g 3015

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 3013	. . 3 |- ( A e. V -> ( [. A / x ]. B = C <-> [_ A / x ]_ B = [_ A / x ]_ C ) )
2	1	notbid 656	. 2 |- ( A e. V -> ( -. [. A / x ]. B = C <-> -. [_ A / x ]_ B = [_ A / x ]_ C ) )
3		df-ne 2307	. . . . 5 |- ( B =/= C <-> -. B = C )
4	3	sbcbii 2963	. . . 4 |- ( [. A / x ]. B =/= C <-> [. A / x ]. -. B = C )
5		sbcng 2944	. . . 4 |- ( A e. V -> ( [. A / x ]. -. B = C <-> -. [. A / x ]. B = C ) )
6	4, 5	syl5bb 191	. . 3 |- ( A e. V -> ( [. A / x ]. B =/= C <-> -. [. A / x ]. B = C ) )
7		df-ne 2307	. . . 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 234	. 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 166	1 |- ( A e. V -> ( [. A / x ]. B =/= C <-> [_ A / x ]_ B =/= [_ A / x ]_ C ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 104   = wceq 1331   e. wcel 1480   =/= wne 2306   [.wsbc 2904   [_csb 2998

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-ne 2307  df-v 2683  df-sbc 2905  df-csb 2999

This theorem is referenced by: (None)