Theorem sbcne12 3986

Description: Distribute proper substitution through an inequality. (Contributed by Andrew Salmon, 18-Jun-2011.) (Revised by NM, 18-Aug-2018.)

Assertion

Ref	Expression
sbcne12	|- ( [. A / x ]. B =/= C <-> [_ A / x ]_ B =/= [_ A / x ]_ C )

Proof of Theorem sbcne12

Step	Hyp	Ref	Expression
1		nne 2798	. . . . . 6 |- ( -. B =/= C <-> B = C )
2	1	sbcbii 3491	. . . . 5 |- ( [. A / x ]. -. B =/= C <-> [. A / x ]. B = C )
3	2	a1i 11	. . . 4 |- ( A e. _V -> ( [. A / x ]. -. B =/= C <-> [. A / x ]. B = C ) )
4		sbcng 3476	. . . 4 |- ( A e. _V -> ( [. A / x ]. -. B =/= C <-> -. [. A / x ]. B =/= C ) )
5		sbceqg 3984	. . . . 5 |- ( A e. _V -> ( [. A / x ]. B = C <-> [_ A / x ]_ B = [_ A / x ]_ C ) )
6		nne 2798	. . . . 5 |- ( -. [_ A / x ]_ B =/= [_ A / x ]_ C <-> [_ A / x ]_ B = [_ A / x ]_ C )
7	5, 6	syl6bbr 278	. . . 4 |- ( A e. _V -> ( [. A / x ]. B = C <-> -. [_ A / x ]_ B =/= [_ A / x ]_ C ) )
8	3, 4, 7	3bitr3d 298	. . 3 |- ( A e. _V -> ( -. [. A / x ]. B =/= C <-> -. [_ A / x ]_ B =/= [_ A / x ]_ C ) )
9	8	con4bid 307	. 2 |- ( A e. _V -> ( [. A / x ]. B =/= C <-> [_ A / x ]_ B =/= [_ A / x ]_ C ) )
10		sbcex 3445	. . . 4 |- ( [. A / x ]. B =/= C -> A e. _V )
11	10	con3i 150	. . 3 |- ( -. A e. _V -> -. [. A / x ]. B =/= C )
12		csbprc 3980	. . . . 5 |- ( -. A e. _V -> [_ A / x ]_ B = (/) )
13		csbprc 3980	. . . . 5 |- ( -. A e. _V -> [_ A / x ]_ C = (/) )
14	12, 13	eqtr4d 2659	. . . 4 |- ( -. A e. _V -> [_ A / x ]_ B = [_ A / x ]_ C )
15	14, 6	sylibr 224	. . 3 |- ( -. A e. _V -> -. [_ A / x ]_ B =/= [_ A / x ]_ C )
16	11, 15	2falsed 366	. 2 |- ( -. A e. _V -> ( [. A / x ]. B =/= C <-> [_ A / x ]_ B =/= [_ A / x ]_ C ) )
17	9, 16	pm2.61i 176	1 |- ( [. A / x ]. B =/= C <-> [_ A / x ]_ B =/= [_ A / x ]_ C )

Syntax hints:   -. wn 3   <-> wb 196   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   [.wsbc 3435   [_csb 3533   (/)c0 3915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-nul 3916

This theorem is referenced by:  disjdsct  29480  cdlemkid3N  36221  cdlemkid4  36222