Theorem bj-vtoclg1fv

Description: Version of bj-vtoclg1f with a disjoint variable condition on x , V . This removes dependency on df-sb and df-clab . Prefer its use over bj-vtoclg1f when sufficient (in particular when V is substituted for _V ). (Contributed by BJ, 14-Sep-2019) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	bj-vtoclg1fv.nf	⊢ Ⅎ 𝑥 𝜓
	bj-vtoclg1fv.maj	⊢ ( 𝑥 = 𝐴 → ( 𝜑 → 𝜓 ) )
	bj-vtoclg1fv.min	⊢ 𝜑
Assertion	bj-vtoclg1fv	⊢ ( 𝐴 ∈ 𝑉 → 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		bj-vtoclg1fv.nf	⊢ Ⅎ 𝑥 𝜓
2		bj-vtoclg1fv.maj	⊢ ( 𝑥 = 𝐴 → ( 𝜑 → 𝜓 ) )
3		bj-vtoclg1fv.min	⊢ 𝜑
4		elissetv	⊢ ( 𝐴 ∈ 𝑉 → ∃ 𝑥 𝑥 = 𝐴 )
5	1 2 3	bj-exlimmpi	⊢ ( ∃ 𝑥 𝑥 = 𝐴 → 𝜓 )
6	4 5	syl	⊢ ( 𝐴 ∈ 𝑉 → 𝜓 )