Theorem bj-ceqsalgv

Description: Version of bj-ceqsalg with a disjoint variable condition on x , V , removing dependency on df-sb and df-clab . Prefer its use over bj-ceqsalg when sufficient (in particular when V is substituted for _V ). (Contributed by BJ, 12-Oct-2019) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	bj-ceqsalgv.1	⊢ Ⅎ 𝑥 𝜓
	bj-ceqsalgv.2	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
Assertion	bj-ceqsalgv	⊢ ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ↔ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		bj-ceqsalgv.1	⊢ Ⅎ 𝑥 𝜓
2		bj-ceqsalgv.2	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
3		elissetv	⊢ ( 𝐴 ∈ 𝑉 → ∃ 𝑥 𝑥 = 𝐴 )
4	1 2	bj-ceqsalg0	⊢ ( ∃ 𝑥 𝑥 = 𝐴 → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ↔ 𝜓 ) )
5	3 4	syl	⊢ ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ↔ 𝜓 ) )