Theorem rspw 3242

Description: Restricted specialization. Weak version of rsp 3253, requiring ax-8 2110, but not ax-12 2178. (Contributed by GG, 3-Oct-2024.)

Hypothesis

Ref	Expression
rspw.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
rspw	⊢ (∀𝑥 ∈ 𝐴 𝜑 → (𝑥 ∈ 𝐴 → 𝜑))

Distinct variable groups:   𝑥,𝑦,𝐴   𝜓,𝑥   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem rspw

Step	Hyp	Ref	Expression
1		df-ral 3068	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
2		eleq1w 2827	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
3		rspw.1	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	2, 3	imbi12d 344	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 → 𝜑) ↔ (𝑦 ∈ 𝐴 → 𝜓)))
5	4	spw 2033	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → (𝑥 ∈ 𝐴 → 𝜑))
6	1, 5	sylbi 217	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (𝑥 ∈ 𝐴 → 𝜑))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1535   ∈ wcel 2108  ∀wral 3067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-clel 2819  df-ral 3068

This theorem is referenced by:  solin  5634