Theorem ceqsralv

Description: Restricted quantifier version of ceqsalv . (Contributed by NM, 21-Jun-2013)

	Ref	Expression
Hypothesis	ceqsralv.2	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
Assertion	ceqsralv	⊢ ( 𝐴 ∈ 𝐵 → ( ∀ 𝑥 ∈ 𝐵 ( 𝑥 = 𝐴 → 𝜑 ) ↔ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		ceqsralv.2	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
2		nfv	⊢ Ⅎ 𝑥 𝜓
3	1	ax-gen	⊢ ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
4		ceqsralt	⊢ ( ( Ⅎ 𝑥 𝜓 ∧ ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) ∧ 𝐴 ∈ 𝐵 ) → ( ∀ 𝑥 ∈ 𝐵 ( 𝑥 = 𝐴 → 𝜑 ) ↔ 𝜓 ) )
5	2 3 4	mp3an12	⊢ ( 𝐴 ∈ 𝐵 → ( ∀ 𝑥 ∈ 𝐵 ( 𝑥 = 𝐴 → 𝜑 ) ↔ 𝜓 ) )