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Metamath Proof Explorer

Theorem prtlem9

Description: Lemma for prter3 . (Contributed by Rodolfo Medina, 25-Sep-2010)

		Ref	Expression
	Assertion	prtlem9	$⊢ A \in B \to \exists x \in B [x] \underset{˙}{\sim} = [A] \underset{˙}{\sim}$

Step	Hyp	Ref	Expression
1		risset	$⊢ A \in B \leftrightarrow \exists x \in B x = A$
2		eceq1	$⊢ x = A \to [x] \underset{˙}{\sim} = [A] \underset{˙}{\sim}$
3	2	reximi	$⊢ \exists x \in B x = A \to \exists x \in B [x] \underset{˙}{\sim} = [A] \underset{˙}{\sim}$
4	1 3	sylbi	$⊢ A \in B \to \exists x \in B [x] \underset{˙}{\sim} = [A] \underset{˙}{\sim}$