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Metamath Proof Explorer

Theorem vtoclf

Description: Implicit substitution of a class for a setvar variable. This is a generalization of chvar . (Contributed by NM, 30-Aug-1993) (Proof shortened by Wolf Lammen, 26-Jan-2025)

	Ref	Expression
Hypotheses	vtoclf.1	$⊢ Ⅎ x ψ$
	vtoclf.2	$⊢ A \in V$
	vtoclf.3	$⊢ x = A \to (φ \leftrightarrow ψ)$
	vtoclf.4	$⊢ φ$
Assertion	vtoclf	$⊢ ψ$

Proof

Step	Hyp	Ref	Expression
1		vtoclf.1	$⊢ Ⅎ x ψ$
2		vtoclf.2	$⊢ A \in V$
3		vtoclf.3	$⊢ x = A \to (φ \leftrightarrow ψ)$
4		vtoclf.4	$⊢ φ$
5	4 3	mpbii	$⊢ x = A \to ψ$
6	1 2 5	vtoclef	$⊢ ψ$