Theorem ceqsralv

Description: Restricted quantifier version of ceqsalv . (Contributed by NM, 21-Jun-2013) Avoid ax-9 , ax-12 , ax-ext . (Revised by SN, 8-Sep-2024)

	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	1	pm5.74i	⊢ ( ( 𝑥 = 𝐴 → 𝜑 ) ↔ ( 𝑥 = 𝐴 → 𝜓 ) )
3	2	ralbii	⊢ ( ∀ 𝑥 ∈ 𝐵 ( 𝑥 = 𝐴 → 𝜑 ) ↔ ∀ 𝑥 ∈ 𝐵 ( 𝑥 = 𝐴 → 𝜓 ) )
4		r19.23v	⊢ ( ∀ 𝑥 ∈ 𝐵 ( 𝑥 = 𝐴 → 𝜓 ) ↔ ( ∃ 𝑥 ∈ 𝐵 𝑥 = 𝐴 → 𝜓 ) )
5		risset	⊢ ( 𝐴 ∈ 𝐵 ↔ ∃ 𝑥 ∈ 𝐵 𝑥 = 𝐴 )
6		pm5.5	⊢ ( ∃ 𝑥 ∈ 𝐵 𝑥 = 𝐴 → ( ( ∃ 𝑥 ∈ 𝐵 𝑥 = 𝐴 → 𝜓 ) ↔ 𝜓 ) )
7	5 6	sylbi	⊢ ( 𝐴 ∈ 𝐵 → ( ( ∃ 𝑥 ∈ 𝐵 𝑥 = 𝐴 → 𝜓 ) ↔ 𝜓 ) )
8	4 7	bitrid	⊢ ( 𝐴 ∈ 𝐵 → ( ∀ 𝑥 ∈ 𝐵 ( 𝑥 = 𝐴 → 𝜓 ) ↔ 𝜓 ) )
9	3 8	bitrid	⊢ ( 𝐴 ∈ 𝐵 → ( ∀ 𝑥 ∈ 𝐵 ( 𝑥 = 𝐴 → 𝜑 ) ↔ 𝜓 ) )