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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 ∈ V
	vtocl.2	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
	vtocl.3	⊢ 𝜑
Assertion	vtocl	⊢ 𝜓

Proof

Step	Hyp	Ref	Expression
1		vtocl.1	⊢ 𝐴 ∈ V
2		vtocl.2	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
3		vtocl.3	⊢ 𝜑
4	3 2	mpbii	⊢ ( 𝑥 = 𝐴 → 𝜓 )
5	1 4	vtocle	⊢ 𝜓