Theorem reueqbidva 49641

Description: Formula-building rule for restricted existential uniqueness quantifier. Deduction form. General version of reueqbidv 3407. (Contributed by Zhi Wang, 20-Nov-2025.)

Hypotheses

Ref	Expression
reueqbidva.1	⊢ (𝜑 → 𝐴 = 𝐵)
reueqbidva.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥

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

Proof of Theorem reueqbidva

Step	Hyp	Ref	Expression
1		reueqbidva.2	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))
2	1	reubidva 3385	. 2 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐴 𝜒))
3		reueqbidva.1	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
4	3	reueqdv 3406	. 2 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜒 ↔ ∃!𝑥 ∈ 𝐵 𝜒))
5	2, 4	bitrd 282	1 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐵 𝜒))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∃!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-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-rex 3092  df-rmo 3371  df-reu 3372

This theorem is used by:  uppropd  50016