Theorem reueqd 3379

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 3377	. 2 ⊢ (𝐴 = 𝐵 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐵 𝜑))
2		rmoeqd.1	. . 3 ⊢ (𝐴 = 𝐵 → (𝜑 ↔ 𝜓))
3	2	reubidv 3361	. 2 ⊢ (𝐴 = 𝐵 → (∃!𝑥 ∈ 𝐵 𝜑 ↔ ∃!𝑥 ∈ 𝐵 𝜓))
4	1, 3	bitrd 280	1 ⊢ (𝐴 = 𝐵 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐵 𝜓))

Syntax hints:   → wi 4   ↔ wb 207   = wceq 1547  ∃!wreu 3343

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-mo 2543  df-eu 2573  df-cleq 2732  df-rex 3065  df-rmo 3345  df-reu 3346

This theorem is referenced by:  aceq1  10037