Theorem reueqdv 3421

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

Hypothesis

Ref	Expression
reueqdv.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

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

Proof of Theorem reueqdv

Step	Hyp	Ref	Expression
1		reueqdv.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		reueq1 3416	. 2 ⊢ (𝐴 = 𝐵 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐵 𝜓))
3	1, 2	syl 17	1 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐵 𝜓))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540  ∃!wreu 3377

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-mo 2539  df-eu 2568  df-cleq 2728  df-rex 3070  df-rmo 3379  df-reu 3380

This theorem is referenced by:  upciclem1  48896