Theorem sbcne12g 3154

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		nne 2520	. . . . 5 |- (-. B =/= C <-> B = C)
2	1	sbcbii 3101	. . . 4 |- ( [.A / x ]. -. B =/= C <-> [.A / x ].B = C)
3	2	a1i 10	. . 3 |- (A e. V -> ( [.A / x ]. -. B =/= C <-> [.A / x ].B = C))
4		sbcng 3086	. . 3 |- (A e. V -> ( [.A / x ]. -. B =/= C <-> -. [.A / x ].B =/= C))
5		sbceqg 3152	. . . 4 |- (A e. V -> ( [.A / x ].B = C <-> [_A / x]_B = [_A / x]_C))
6		nne 2520	. . . 4 |- (-. [_A / x]_B =/= [_A / x]_C <-> [_A / x]_B = [_A / x]_C)
7	5, 6	syl6bbr 254	. . 3 |- (A e. V -> ( [.A / x ].B = C <-> -. [_A / x]_B =/= [_A / x]_C))
8	3, 4, 7	3bitr3d 274	. 2 |- (A e. V -> (-. [.A / x ].B =/= C <-> -. [_A / x]_B =/= [_A / x]_C))
9	8	con4bid 284	1 |- (A e. V -> ( [.A / x ].B =/= C <-> [_A / x]_B =/= [_A / x]_C))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   = wceq 1642   e. wcel 1710   =/= wne 2516   [.wsbc 3046  [_csb 3136

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-v 2861  df-sbc 3047  df-csb 3137

This theorem is referenced by: (None)