Theorem rsp3 38965

Description: From a restricted universal statement over 𝐴, specialize to an arbitrary element 𝑦 ∈ 𝐴, cf. rsp 3260. (Contributed by Peter Mazsa, 9-Feb-2026.)

Hypotheses

Ref	Expression
rsp3.1	⊢ Ⅎ𝑥𝐴
rsp3.2	⊢ Ⅎ𝑦𝐴
rsp3.3	⊢ Ⅎ𝑦𝜑
rsp3.4	⊢ Ⅎ𝑥𝜓
rsp3.5	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
rsp3	⊢ (∀𝑥 ∈ 𝐴 𝜑 → (𝑦 ∈ 𝐴 → 𝜓))

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem rsp3

Step	Hyp	Ref	Expression
1		rsp3.1	. . 3 ⊢ Ⅎ𝑥𝐴
2		rsp3.2	. . 3 ⊢ Ⅎ𝑦𝐴
3		rsp3.3	. . 3 ⊢ Ⅎ𝑦𝜑
4		rsp3.4	. . 3 ⊢ Ⅎ𝑥𝜓
5		rsp3.5	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
6	1, 2, 3, 4, 5	cbvralfw 3312	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓)
7		rsp 3260	. 2 ⊢ (∀𝑦 ∈ 𝐴 𝜓 → (𝑦 ∈ 𝐴 → 𝜓))
8	6, 7	sylbi 220	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (𝑦 ∈ 𝐴 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 209  Ⅎwnf 1811   ∈ wcel 2150  Ⅎwnfc 2917  ∀wral 3086

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-11 2199  ax-12 2220

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1808  df-nf 1812  df-clel 2845  df-nfc 2919  df-ral 3087

This theorem is referenced by:  rsp3eq  38966