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Metamath Proof Explorer

Theorem vtoclg

Description: Implicit substitution of a class expression for a setvar variable. (Contributed by NM, 17-Apr-1995) Avoid ax-12 . (Revised by SN, 20-Apr-2024) (Proof shortened by Wolf Lammen, 26-Jan-2025)

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

Proof

Step	Hyp	Ref	Expression
1		vtoclg.1	$⊢ x = A \to (φ \leftrightarrow ψ)$
2		vtoclg.2	$⊢ φ$
3	2 1	mpbii	$⊢ x = A \to ψ$
4	3	vtocleg	$⊢ A \in V \to ψ$