Theorem sbcne12g 3145

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 3143	. . 3 |- ( A e. V -> ( [. A / x ]. B = C <-> [_ A / x ]_ B = [_ A / x ]_ C ) )
2	1	notbid 673	. 2 |- ( A e. V -> ( -. [. A / x ]. B = C <-> -. [_ A / x ]_ B = [_ A / x ]_ C ) )
3		df-ne 2403	. . . . 5 |- ( B =/= C <-> -. B = C )
4	3	sbcbii 3091	. . . 4 |- ( [. A / x ]. B =/= C <-> [. A / x ]. -. B = C )
5		sbcng 3072	. . . 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 2403	. . . 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 1397   e. wcel 2202   =/= wne 2402   [.wsbc 3031   [_csb 3127

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-v 2804  df-sbc 3032  df-csb 3128

This theorem is referenced by: (None)