Theorem vtoclri

Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 21-Nov-1994)

	Ref	Expression
Hypotheses	vtoclri.1	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
	vtoclri.2	⊢ ∀ 𝑥 ∈ 𝐵 𝜑
Assertion	vtoclri	⊢ ( 𝐴 ∈ 𝐵 → 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		vtoclri.1	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
2		vtoclri.2	⊢ ∀ 𝑥 ∈ 𝐵 𝜑
3	2	rspec	⊢ ( 𝑥 ∈ 𝐵 → 𝜑 )
4	1 3	vtoclga	⊢ ( 𝐴 ∈ 𝐵 → 𝜓 )