Theorem ssofeq 4077

Description: When A and B are subsets of C, equality depends only on the elements of C. (Contributed by SF, 13-Jan-2015.)

Assertion

Ref	Expression
ssofeq	|- ((A C_ C /\ B C_ C) -> (A = B <-> A.x e. C (x e. A <-> x e. B)))

Distinct variable groups:   x,A   x,B   x,C

Proof of Theorem ssofeq

Step	Hyp	Ref	Expression
1		ssofss 4076	. . 3 |- (A C_ C -> (A C_ B <-> A.x e. C (x e. A -> x e. B)))
2		ssofss 4076	. . 3 |- (B C_ C -> (B C_ A <-> A.x e. C (x e. B -> x e. A)))
3	1, 2	bi2anan9 843	. 2 |- ((A C_ C /\ B C_ C) -> ((A C_ B /\ B C_ A) <-> (A.x e. C (x e. A -> x e. B) /\ A.x e. C (x e. B -> x e. A))))
4		eqss 3287	. 2 |- (A = B <-> (A C_ B /\ B C_ A))
5		ralbiim 2751	. 2 |- (A.x e. C (x e. A <-> x e. B) <-> (A.x e. C (x e. A -> x e. B) /\ A.x e. C (x e. B -> x e. A)))
6	3, 4, 5	3bitr4g 279	1 |- ((A C_ C /\ B C_ C) -> (A = B <-> A.x e. C (x e. A <-> x e. B)))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   = wceq 1642   e. wcel 1710  A.wral 2614   C_ wss 3257

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-ss 3259

This theorem is referenced by:  eqpw1  4162  pw111  4170  eqrelk  4212  sikexlem  4295  insklem  4304