Theorem ssdif 2162

Description: Difference law for subsets.

Assertion

Ref	Expression
ssdif	|- (A (_ B -> (A \ C) (_ (B \ C))

Proof of Theorem ssdif

Step	Hyp	Ref	Expression
1		ssel 2053	. . . 4 |- (A (_ B -> (x e. A -> x e. B))
2	1	anim1d 558	. . 3 |- (A (_ B -> ((x e. A /\ -. x e. C) -> (x e. B /\ -. x e. C)))
3		eldif 2047	. . 3 |- (x e. (A \ C) <-> (x e. A /\ -. x e. C))
4		eldif 2047	. . 3 |- (x e. (B \ C) <-> (x e. B /\ -. x e. C))
5	2, 3, 4	3imtr4g 551	. 2 |- (A (_ B -> (x e. (A \ C) -> x e. (B \ C)))
6	5	ssrdv 2060	1 |- (A (_ B -> (A \ C) (_ (B \ C))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   e. wcel 955   \ cdif 2034   (_ wss 2037

This theorem is referenced by:  sspr 2466  php 4493  pssnn 4513

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803  df-dif 2039  df-in 2041  df-ss 2043

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