Theorem rsp3eq 38749

Description: From a restricted universal statement over 𝐴, specialize to an arbitrary element class, cf. rsp3 38748. (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
rsp3eq	⊢ (∀𝑥 ∈ 𝐴 𝜑 → ((𝑦 = 𝐵 ∧ 𝐵 ∈ 𝐴) → 𝜓))

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem rsp3eq

Step	Hyp	Ref	Expression
1		eqeltr 38622	. 2 ⊢ ((𝑦 = 𝐵 ∧ 𝐵 ∈ 𝐴) → 𝑦 ∈ 𝐴)
2		rsp3.1	. . 3 ⊢ Ⅎ𝑥𝐴
3		rsp3.2	. . 3 ⊢ Ⅎ𝑦𝐴
4		rsp3.3	. . 3 ⊢ Ⅎ𝑦𝜑
5		rsp3.4	. . 3 ⊢ Ⅎ𝑥𝜓
6		rsp3.5	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
7	2, 3, 4, 5, 6	rsp3 38748	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (𝑦 ∈ 𝐴 → 𝜓))
8	1, 7	syl5 34	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ((𝑦 = 𝐵 ∧ 𝐵 ∈ 𝐴) → 𝜓))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   = wceq 1548  Ⅎwnf 1791   ∈ wcel 2121  Ⅎwnfc 2888  ∀wral 3055

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-11 2170  ax-12 2191  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-ex 1788  df-nf 1792  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ral 3056

This theorem is referenced by: (None)