This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem csbieb

Description: Bidirectional conversion between an implicit class substitution hypothesis x = A -> B = C and its explicit substitution equivalent. (Contributed by NM, 2-Mar-2008)

	Ref	Expression
Hypotheses	csbieb.1	$⊢ A \in V$
	csbieb.2	$⊢ \underline{Ⅎ} x C$
Assertion	csbieb	$⊢ \forall x (x = A \to B = C) \leftrightarrow ⦋ A / x ⦌ B = C$

Proof

Step	Hyp	Ref	Expression
1		csbieb.1	$⊢ A \in V$
2		csbieb.2	$⊢ \underline{Ⅎ} x C$
3		csbiebt	$⊢ (A \in V \land \underline{Ⅎ} x C) \to (\forall x (x = A \to B = C) \leftrightarrow ⦋ A / x ⦌ B = C)$
4	1 2 3	mp2an	$⊢ \forall x (x = A \to B = C) \leftrightarrow ⦋ A / x ⦌ B = C$