Theorem reueqbidva 49165

Description: Formula-building rule for restricted existential uniqueness quantifier. Deduction form. General version of reueqbidv 3390. (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 3366	. 2 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐴 𝜒))
3		reueqbidva.1	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
4	3	reueqdv 3389	. 2 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜒 ↔ ∃!𝑥 ∈ 𝐵 𝜒))
5	2, 4	bitrd 279	1 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐵 𝜒))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∃!wreu 3350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-mo 2540  df-eu 2570  df-cleq 2729  df-rex 3063  df-rmo 3352  df-reu 3353

This theorem is referenced by:  uppropd  49540