Theorem ssextss

Description: An extensionality-like principle defining subclass in terms of subsets. (Contributed by NM, 30-Jun-2004)

	Ref	Expression
Assertion	ssextss	|- ( A C_ B <-> A. x ( x C_ A -> x C_ B ) )

Proof

Step	Hyp	Ref	Expression
1		sspwb	|- ( A C_ B <-> ~P A C_ ~P B )
2		df-ss	|- ( ~P A C_ ~P B <-> A. x ( x e. ~P A -> x e. ~P B ) )
3		velpw	|- ( x e. ~P A <-> x C_ A )
4		velpw	|- ( x e. ~P B <-> x C_ B )
5	3 4	imbi12i	|- ( ( x e. ~P A -> x e. ~P B ) <-> ( x C_ A -> x C_ B ) )
6	5	albii	|- ( A. x ( x e. ~P A -> x e. ~P B ) <-> A. x ( x C_ A -> x C_ B ) )
7	1 2 6	3bitri	|- ( A C_ B <-> A. x ( x C_ A -> x C_ B ) )