Theorem sbcnel12g 3154

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

Assertion

Ref	Expression
sbcnel12g	|- (A e. V -> ( [.A / x ].B e/ C <-> [_A / x]_B e/ [_A / x]_C))

Proof of Theorem sbcnel12g

Step	Hyp	Ref	Expression
1		df-nel 2520	. . . 4 |- (B e/ C <-> -. B e. C)
2	1	sbcbii 3102	. . 3 |- ( [.A / x ].B e/ C <-> [.A / x ]. -. B e. C)
3	2	a1i 10	. 2 |- (A e. V -> ( [.A / x ].B e/ C <-> [.A / x ]. -. B e. C))
4		sbcng 3087	. 2 |- (A e. V -> ( [.A / x ]. -. B e. C <-> -. [.A / x ].B e. C))
5		sbcel12g 3152	. . . 4 |- (A e. V -> ( [.A / x ].B e. C <-> [_A / x]_B e. [_A / x]_C))
6	5	notbid 285	. . 3 |- (A e. V -> (-. [.A / x ].B e. C <-> -. [_A / x]_B e. [_A / x]_C))
7		df-nel 2520	. . 3 |- ([_A / x]_B e/ [_A / x]_C <-> -. [_A / x]_B e. [_A / x]_C)
8	6, 7	syl6bbr 254	. 2 |- (A e. V -> (-. [.A / x ].B e. C <-> [_A / x]_B e/ [_A / x]_C))
9	3, 4, 8	3bitrd 270	1 |- (A e. V -> ( [.A / x ].B e/ C <-> [_A / x]_B e/ [_A / x]_C))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   e. wcel 1710   e/ wnel 2518   [.wsbc 3047  [_csb 3137

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 2479  df-nel 2520  df-v 2862  df-sbc 3048  df-csb 3138

This theorem is referenced by: (None)