Theorem vtoclgOLD

Description: Obsolete version of vtoclg as of 26-Jan-2025. (Contributed by NM, 17-Apr-1995) Avoid ax-12 . (Revised by SN, 20-Apr-2024) (New usage is discouraged.) (Proof modification is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		vtoclgOLD.1	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
2		vtoclgOLD.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 )