Theorem reueqbidv 3407

Description: Formula-building rule for restricted existential uniqueness quantifier. Deduction form. General version of reubidv 3387. (Contributed by GG, 1-Sep-2025.)

Hypotheses

Ref	Expression
reueqbidv.1	⊢ (𝜑 → 𝐴 = 𝐵)
reueqbidv.2	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
reueqbidv	⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐵 𝜒))

Distinct variable group:   𝜑,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem reueqbidv

Step	Hyp	Ref	Expression
1		reueqbidv.1	. . . . 5 ⊢ (𝜑 → 𝐴 = 𝐵)
2	1	eleq2d 2851	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
3		reueqbidv.2	. . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
4	2, 3	anbi12d 644	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒)))
5	4	eubidv 2616	. 2 ⊢ (𝜑 → (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜒)))
6		df-reu 3372	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
7		df-reu 3372	. 2 ⊢ (∃!𝑥 ∈ 𝐵 𝜒 ↔ ∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜒))
8	5, 6, 7	3bitr4g 317	1 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐵 𝜒))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ 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  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-mo 2569  df-eu 2599  df-cleq 2757  df-clel 2840  df-reu 3372

This theorem is used by:  upciclem1  50001  upfval  50011  isuplem  50014  upeu3  50030  oppcup3lem  50041  oppcup  50042