Theorem reueqd 3344

Description: Equality deduction for restricted unique existential quantifier. (Contributed by NM, 5-Apr-2004.)

Hypothesis

Ref	Expression
raleqd.1	⊢ (𝐴 = 𝐵 → (𝜑 ↔ 𝜓))

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

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

Proof of Theorem reueqd

Step	Hyp	Ref	Expression
1		reueq1 3332	. 2 ⊢ (𝐴 = 𝐵 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐵 𝜑))
2		raleqd.1	. . 3 ⊢ (𝐴 = 𝐵 → (𝜑 ↔ 𝜓))
3	2	reubidv 3314	. 2 ⊢ (𝐴 = 𝐵 → (∃!𝑥 ∈ 𝐵 𝜑 ↔ ∃!𝑥 ∈ 𝐵 𝜓))
4	1, 3	bitrd 271	1 ⊢ (𝐴 = 𝐵 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐵 𝜓))

Syntax hints:   → wi 4   ↔ wb 198   = wceq 1601  ∃!wreu 3092

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-ext 2754

This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1824  df-mo 2551  df-eu 2587  df-cleq 2770  df-clel 2774  df-reu 3097

This theorem is referenced by:  aceq1  9273