Theorem ssconb 2166

Description: Contraposition law for subsets.

Assertion

Ref	Expression
ssconb	|- ((A (_ C /\ B (_ C) -> (A (_ (C \ B) <-> B (_ (C \ A)))

Proof of Theorem ssconb

Step	Hyp	Ref	Expression
1		pm5.1 675	. . . . . . 7 |- (((x e. A -> x e. C) /\ (x e. B -> x e. C)) -> ((x e. A -> x e. C) <-> (x e. B -> x e. C)))
2		ssel 2059	. . . . . . 7 |- (A (_ C -> (x e. A -> x e. C))
3		ssel 2059	. . . . . . 7 |- (B (_ C -> (x e. B -> x e. C))
4	1, 2, 3	syl2an 454	. . . . . 6 |- ((A (_ C /\ B (_ C) -> ((x e. A -> x e. C) <-> (x e. B -> x e. C)))
5		bi2.03 165	. . . . . . 7 |- ((x e. A -> -. x e. B) <-> (x e. B -> -. x e. A))
6	5	a1i 8	. . . . . 6 |- ((A (_ C /\ B (_ C) -> ((x e. A -> -. x e. B) <-> (x e. B -> -. x e. A)))
7	4, 6	anbi12d 627	. . . . 5 |- ((A (_ C /\ B (_ C) -> (((x e. A -> x e. C) /\ (x e. A -> -. x e. B)) <-> ((x e. B -> x e. C) /\ (x e. B -> -. x e. A))))
8		jcab 597	. . . . 5 |- ((x e. A -> (x e. C /\ -. x e. B)) <-> ((x e. A -> x e. C) /\ (x e. A -> -. x e. B)))
9		jcab 597	. . . . 5 |- ((x e. B -> (x e. C /\ -. x e. A)) <-> ((x e. B -> x e. C) /\ (x e. B -> -. x e. A)))
10	7, 8, 9	3bitr4g 554	. . . 4 |- ((A (_ C /\ B (_ C) -> ((x e. A -> (x e. C /\ -. x e. B)) <-> (x e. B -> (x e. C /\ -. x e. A))))
11		eldif 2053	. . . . 5 |- (x e. (C \ B) <-> (x e. C /\ -. x e. B))
12	11	imbi2i 185	. . . 4 |- ((x e. A -> x e. (C \ B)) <-> (x e. A -> (x e. C /\ -. x e. B)))
13		eldif 2053	. . . . 5 |- (x e. (C \ A) <-> (x e. C /\ -. x e. A))
14	13	imbi2i 185	. . . 4 |- ((x e. B -> x e. (C \ A)) <-> (x e. B -> (x e. C /\ -. x e. A)))
15	10, 12, 14	3bitr4g 554	. . 3 |- ((A (_ C /\ B (_ C) -> ((x e. A -> x e. (C \ B)) <-> (x e. B -> x e. (C \ A))))
16	15	albidv 1276	. 2 |- ((A (_ C /\ B (_ C) -> (A.x(x e. A -> x e. (C \ B)) <-> A.x(x e. B -> x e. (C \ A))))
17		dfss2 2054	. 2 |- (A (_ (C \ B) <-> A.x(x e. A -> x e. (C \ B)))
18		dfss2 2054	. 2 |- (B (_ (C \ A) <-> A.x(x e. B -> x e. (C \ A)))
19	16, 17, 18	3bitr4g 554	1 |- ((A (_ C /\ B (_ C) -> (A (_ (C \ B) <-> B (_ (C \ A)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223  A.wal 952   e. wcel 956   \ cdif 2040   (_ wss 2043

This theorem is referenced by:  sbthlem1 4433  sbthlem2 4434  clsval2 7635

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-or 224  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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