Theorem cbvreudavw 36822

Description: Change bound variable in the restricted existential uniqueness quantifier. Deduction form. (Contributed by GG, 14-Aug-2025.)

Hypothesis

Ref	Expression
cbvreudavw.1	⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
cbvreudavw	⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑦 ∈ 𝐴 𝜒))

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

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

Proof of Theorem cbvreudavw

Step	Hyp	Ref	Expression
1		eleq1w 2848	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
2	1	adantl 487	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
3		cbvreudavw.1	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))
4	2, 3	anbi12d 644	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑦 ∈ 𝐴 ∧ 𝜒)))
5	4	cbveudavw 36820	. 2 ⊢ (𝜑 → (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜒)))
6		df-reu 3372	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
7		df-reu 3372	. 2 ⊢ (∃!𝑦 ∈ 𝐴 𝜒 ↔ ∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜒))
8	5, 6, 7	3bitr4g 317	1 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑦 ∈ 𝐴 𝜒))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2146  ∃!weu 2598  ∃!wreu 3369

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-mo 2569  df-eu 2599  df-clel 2840  df-reu 3372

This theorem is used by: (None)