Theorem vtoclg

Description: Implicit substitution of a class expression for a setvar variable. (Contributed by NM, 17-Apr-1995) Avoid ax-12 . (Revised by SN, 20-Apr-2024)

	Ref	Expression
Hypotheses	vtoclg.1	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
	vtoclg.2	⊢ 𝜑
Assertion	vtoclg	⊢ ( 𝐴 ∈ 𝑉 → 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		vtoclg.1	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
2		vtoclg.2	⊢ 𝜑
3		elisset	⊢ ( 𝐴 ∈ 𝑉 → ∃ 𝑥 𝑥 = 𝐴 )
4	2 1	mpbii	⊢ ( 𝑥 = 𝐴 → 𝜓 )
5	4	exlimiv	⊢ ( ∃ 𝑥 𝑥 = 𝐴 → 𝜓 )
6	3 5	syl	⊢ ( 𝐴 ∈ 𝑉 → 𝜓 )