Theorem ssconb 1598

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		pm5.1 501	. . . . . . 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 1502	. . . . . . 7 |- (A (_ C -> (x e. A -> x e. C))
3		ssel 1502	. . . . . . 7 |- (B (_ C -> (x e. B -> x e. C))
4	1, 2, 3	syl2an 349	. . . . . 6 |- ((A (_ C /\ B (_ C) -> ((x e. A -> x e. C) <-> (x e. B -> x e. C)))
5		bi2.03 144	. . . . . . 7 |- ((x e. A -> -. x e. B) <-> (x e. B -> -. x e. A))
6	5	a1i 7	. . . . . 6 |- ((A (_ C /\ B (_ C) -> ((x e. A -> -. x e. B) <-> (x e. B -> -. x e. A)))
7	4, 6	anbi12d 476	. . . . 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 454	. . . . 5 |- ((x e. A -> (x e. C /\ -. x e. B)) <-> ((x e. A -> x e. C) /\ (x e. A -> -. x e. B)))
9		jcab 454	. . . . 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 428	. . . 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 1496	. . . . 5 |- (x e. (C \ B) <-> (x e. C /\ -. x e. B))
12	11	imbi2i 160	. . . 4 |- ((x e. A -> x e. (C \ B)) <-> (x e. A -> (x e. C /\ -. x e. B)))
13		eldif 1496	. . . . 5 |- (x e. (C \ A) <-> (x e. C /\ -. x e. A))
14	13	imbi2i 160	. . . 4 |- ((x e. B -> x e. (C \ A)) <-> (x e. B -> (x e. C /\ -. x e. A)))
15	10, 12, 14	3bitr4g 428	. . 3 |- ((A (_ C /\ B (_ C) -> ((x e. A -> x e. (C \ B)) <-> (x e. B -> x e. (C \ A))))
16	15	bialdv 935	. 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 1497	. 2 |- (A (_ (C \ B) <-> A.x(x e. A -> x e. (C \ B)))
18		dfss2 1497	. 2 |- (B (_ (C \ A) <-> A.x(x e. B -> x e. (C \ A)))
19	16, 17, 18	3bitr4g 428	1 |- ((A (_ C /\ B (_ C) -> (A (_ (C \ B) <-> B (_ (C \ A)))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672   e. wcel 1092   \ cdif 1484   (_ wss 1487

This theorem is referenced by:  sbthlem1 3349  sbthlem2 3350

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-in 1491  df-ss 1492

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