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Metamath Proof Explorer

Theorem vtoclga

Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 20-Aug-1995) Avoid ax-10 and ax-11 . (Revised by GG, 20-Aug-2023)

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

Proof

Step	Hyp	Ref	Expression
1		vtoclga.1	$⊢ x = A \to (φ \leftrightarrow ψ)$
2		vtoclga.2	$⊢ x \in B \to φ$
3		eleq1	$⊢ x = A \to (x \in B \leftrightarrow A \in B)$
4	3 1	imbi12d	$⊢ x = A \to ((x \in B \to φ) \leftrightarrow (A \in B \to ψ))$
5	4 2	vtoclg	$⊢ A \in B \to (A \in B \to ψ)$
6	5	pm2.43i	$⊢ A \in B \to ψ$