Theorem psseq12d

Description: An equality deduction for the proper subclass relationship. (Contributed by NM, 9-Jun-2004)

	Ref	Expression
Hypotheses	psseq1d.1	$$⊢\phi \to A=B$$
	psseq12d.2	$$⊢\phi \to C=D$$
Assertion	psseq12d	$$⊢\phi \to \left(A\subset C\leftrightarrow B\subset D\right)$$

Proof

Step	Hyp	Ref	Expression
1		psseq1d.1	$$⊢\phi \to A=B$$
2		psseq12d.2	$$⊢\phi \to C=D$$
3	1	psseq1d	$$⊢\phi \to \left(A\subset C\leftrightarrow B\subset C\right)$$
4	2	psseq2d	$$⊢\phi \to \left(B\subset C\leftrightarrow B\subset D\right)$$
5	3 4	bitrd	$$⊢\phi \to \left(A\subset C\leftrightarrow B\subset D\right)$$