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

Metamath Proof Explorer

Theorem vtocl

Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 30-Aug-1993) Remove dependency on ax-10 . (Revised by BJ, 29-Nov-2020) (Proof shortened by SN, 20-Apr-2024) (Proof shortened by Wolf Lammen, 20-Jun-2025)

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

Proof

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