Theorem reueqd 3421

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

Hypothesis

Ref	Expression
rmoeqd.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 3417	. 2 ⊢ (𝐴 = 𝐵 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐵 𝜑))
2		rmoeqd.1	. . 3 ⊢ (𝐴 = 𝐵 → (𝜑 ↔ 𝜓))
3	2	reubidv 3398	. 2 ⊢ (𝐴 = 𝐵 → (∃!𝑥 ∈ 𝐵 𝜑 ↔ ∃!𝑥 ∈ 𝐵 𝜓))
4	1, 3	bitrd 279	1 ⊢ (𝐴 = 𝐵 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐵 𝜓))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540  ∃!wreu 3378

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 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-mo 2540  df-eu 2569  df-cleq 2729  df-rex 3071  df-rmo 3380  df-reu 3381

This theorem is referenced by:  aceq1  10157