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Metamath Proof Explorer

Theorem freq2

Description: Equality theorem for the well-founded predicate. (Contributed by NM, 3-Apr-1994)

		Ref	Expression
	Assertion	freq2	$⊢ A = B \to (R Fr A \leftrightarrow R Fr B)$

Step	Hyp	Ref	Expression
1		eqimss2	$⊢ A = B \to B \subseteq A$
2		frss	$⊢ B \subseteq A \to (R Fr A \to R Fr B)$
3	1 2	syl	$⊢ A = B \to (R Fr A \to R Fr B)$
4		eqimss	$⊢ A = B \to A \subseteq B$
5		frss	$⊢ A \subseteq B \to (R Fr B \to R Fr A)$
6	4 5	syl	$⊢ A = B \to (R Fr B \to R Fr A)$
7	3 6	impbid	$⊢ A = B \to (R Fr A \leftrightarrow R Fr B)$