Theorem ssdif 2168

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 2059	. . . 4 |- (A (_ B -> (x e. A -> x e. B))
2	1	anim1d 559	. . 3 |- (A (_ B -> ((x e. A /\ -. x e. C) -> (x e. B /\ -. x e. C)))
3		eldif 2053	. . 3 |- (x e. (A \ C) <-> (x e. A /\ -. x e. C))
4		eldif 2053	. . 3 |- (x e. (B \ C) <-> (x e. B /\ -. x e. C))
5	2, 3, 4	3imtr4g 552	. 2 |- (A (_ B -> (x e. (A \ C) -> x e. (B \ C)))
6	5	ssrdv 2066	1 |- (A (_ B -> (A \ C) (_ (B \ C))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   e. wcel 956   \ cdif 2040   (_ wss 2043

This theorem is referenced by:  sspr 2471  php 4499  pssnn 4519

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808  df-dif 2045  df-in 2047  df-ss 2049

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