Theorem cbvreuvw 3361

Description: Change the bound variable of a restricted unique existential quantifier using implicit substitution. Version of cbvreuv 3365 with a disjoint variable condition, which requires fewer axioms. (Contributed by NM, 5-Apr-2004.) (Revised by Gino Giotto, 30-Sep-2024.)

Hypothesis

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

Assertion

Ref	Expression
cbvreuvw	⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓)

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

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

Proof of Theorem cbvreuvw

Step	Hyp	Ref	Expression
1		eleq1w 2820	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
2		cbvralvw.1	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
3	1, 2	anbi12d 634	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑦 ∈ 𝐴 ∧ 𝜓)))
4	3	cbveuvw 2605	. 2 ⊢ (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
5		df-reu 3068	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
6		df-reu 3068	. 2 ⊢ (∃!𝑦 ∈ 𝐴 𝜓 ↔ ∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
7	4, 5, 6	3bitr4i 306	1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑦 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∈ wcel 2110  ∃!weu 2567  ∃!wreu 3063

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 1976  ax-7 2016  ax-8 2112

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788  df-mo 2539  df-eu 2568  df-clel 2816  df-reu 3068

This theorem is referenced by:  reu8  3646  aceq1  9731  aceq2  9733  fin23lem27  9942  divalglem10  15963  lspsneu  20160  lshpsmreu  36860  wessf1ornlem  42395  fourierdlem50  43372