Theorem reueqdv 3404

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 3401	. 2 ⊢ (𝐴 = 𝐵 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐵 𝜓))
3	1, 2	syl 18	1 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐵 𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570  ∃!wreu 3367

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 1945  ax-6 2002  ax-7 2043  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-mo 2567  df-eu 2597  df-cleq 2755  df-rex 3091  df-rmo 3369  df-reu 3370

This theorem is used by:  reueqbidva  50119