Theorem reueqbidv 3380

Description: Formula-building rule for restricted existential uniqueness quantifier. Deduction form. General version of reubidv 3360. (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 2825	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
3		reueqbidv.2	. . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
4	2, 3	anbi12d 638	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ 𝜒)))
5	4	eubidv 2590	. 2 ⊢ (𝜑 → (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜒)))
6		df-reu 3345	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
7		df-reu 3345	. 2 ⊢ (∃!𝑥 ∈ 𝐵 𝜒 ↔ ∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜒))
8	5, 6, 7	3bitr4g 315	1 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐵 𝜒))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∃!weu 2572  ∃!wreu 3342

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-mo 2543  df-eu 2573  df-cleq 2731  df-clel 2814  df-reu 3345

This theorem is referenced by:  upciclem1  49656  upfval  49666  isuplem  49669  upeu3  49685  oppcup3lem  49696  oppcup  49697