Theorem rspw

Description: Restricted specialization. Weak version of rsp , requiring ax-8 , but not ax-12 . (Contributed by Gino Giotto, 3-Oct-2024)

	Ref	Expression
Hypothesis	rspw.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
Assertion	rspw	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 → ( 𝑥 ∈ 𝐴 → 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		rspw.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
2		df-ral	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) )
3		eleq1w	⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴 ) )
4	3 1	imbi12d	⊢ ( 𝑥 = 𝑦 → ( ( 𝑥 ∈ 𝐴 → 𝜑 ) ↔ ( 𝑦 ∈ 𝐴 → 𝜓 ) ) )
5	4	spw	⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) → ( 𝑥 ∈ 𝐴 → 𝜑 ) )
6	2 5	sylbi	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 → ( 𝑥 ∈ 𝐴 → 𝜑 ) )