Theorem reueqbidv 3412

Description: Formula-building rule for restricted existential uniqueness quantifier. Deduction form. General version of reubidv 3392. (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 2855	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
3		reueqbidv.2	. . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
4	2, 3	anbi12d 643	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒)))
5	4	eubidv 2620	. 2 ⊢ (𝜑 → (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜒)))
6		df-reu 3377	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
7		df-reu 3377	. 2 ⊢ (∃!𝑥 ∈ 𝐵 𝜒 ↔ ∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜒))
8	5, 6, 7	3bitr4g 317	1 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐵 𝜒))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∃!weu 2602  ∃!wreu 3374

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-mo 2573  df-eu 2603  df-cleq 2761  df-clel 2844  df-reu 3377

This theorem is referenced by:  upciclem1  49824  upfval  49834  isuplem  49837  upeu3  49853  oppcup3lem  49864  oppcup  49865