Theorem eqssd

Description: Equality deduction from two subclass relationships. Compare Theorem 4 of Suppes p. 22. (Contributed by NM, 27-Jun-2004)

	Ref	Expression
Hypotheses	eqssd.1	$$⊢\phi \to A\subseteq B$$
	eqssd.2	$$⊢\phi \to B\subseteq A$$
Assertion	eqssd	$$⊢\phi \to A=B$$

Proof

Step	Hyp	Ref	Expression
1		eqssd.1	$$⊢\phi \to A\subseteq B$$
2		eqssd.2	$$⊢\phi \to B\subseteq A$$
3		eqss	$$⊢A=B\leftrightarrow \left(A\subseteq B\wedge B\subseteq A\right)$$
4	1 2 3	sylanbrc	$$⊢\phi \to A=B$$