Theorem vtoclefex

Description: Implicit substitution of a class for a setvar variable. (Contributed by ML, 17-Oct-2020)

	Ref	Expression
Hypotheses	vtoclefex.1	⊢ Ⅎ 𝑥 𝜑
	vtoclefex.3	⊢ ( 𝑥 = 𝐴 → 𝜑 )
Assertion	vtoclefex	⊢ ( 𝐴 ∈ 𝑉 → 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		vtoclefex.1	⊢ Ⅎ 𝑥 𝜑
2		vtoclefex.3	⊢ ( 𝑥 = 𝐴 → 𝜑 )
3	2	ax-gen	⊢ ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 )
4		vtoclegft	⊢ ( ( 𝐴 ∈ 𝑉 ∧ Ⅎ 𝑥 𝜑 ∧ ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ) → 𝜑 )
5	1 3 4	mp3an23	⊢ ( 𝐴 ∈ 𝑉 → 𝜑 )