Theorem sbcnel12g 16408

Description: Distribute proper substitution through negated membership.

Assertion

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

Proof of Theorem sbcnel12g

Step	Hyp	Ref	Expression
1		df-nel 2020	. . 3 |- (B e/ C <-> -. B e. C)
2	1	sbcbii 2506	. 2 |- (A e. D -> ([A / x]B e/ C <-> [A / x] -. B e. C))
3		sbcng 2495	. 2 |- (A e. D -> ([A / x] -. B e. C <-> -. [A / x]B e. C))
4		sbcel12g 2552	. . . 4 |- (A e. D -> ([A / x]B e. C <-> [_A / x]_B e. [_A / x]_C))
5	4	notbid 673	. . 3 |- (A e. D -> (-. [A / x]B e. C <-> -. [_A / x]_B e. [_A / x]_C))
6		df-nel 2020	. . 3 |- ([_A / x]_B e/ [_A / x]_C <-> -. [_A / x]_B e. [_A / x]_C)
7	5, 6	syl6bbr 597	. 2 |- (A e. D -> (-. [A / x]B e. C <-> [_A / x]_B e/ [_A / x]_C))
8	2, 3, 7	3bitrd 603	1 |- (A e. D -> ([A / x]B e/ C <-> [_A / x]_B e/ [_A / x]_C))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   e. wcel 1300  [wsbc 1534   e/ wnel 2018  [_csb 2540

This theorem is referenced by:  rusbcALT 16410

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-nel 2020  df-v 2294  df-sbc 2454  df-csb 2541

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