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Metamath Proof Explorer

Theorem hbxfrbi

Description: A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfreq for equality version. (Contributed by Jonathan Ben-Naim, 3-Jun-2011)

	Ref	Expression
Hypotheses	hbxfrbi.1	$⊢ φ \leftrightarrow ψ$
	hbxfrbi.2	$⊢ ψ \to \forall x ψ$
Assertion	hbxfrbi	$⊢ φ \to \forall x φ$

Proof

Step	Hyp	Ref	Expression
1		hbxfrbi.1	$⊢ φ \leftrightarrow ψ$
2		hbxfrbi.2	$⊢ ψ \to \forall x ψ$
3	1	albii	$⊢ \forall x φ \leftrightarrow \forall x ψ$
4	2 1 3	3imtr4i	$⊢ φ \to \forall x φ$