Theorem 2reurmo 3724

Description: Double restricted quantification with restricted existential uniqueness and restricted "at most one", analogous to 2eumo 2672. (Contributed by Alexander van der Vekens, 24-Jun-2017.)

Assertion

Ref	Expression
2reurmo	⊢ (∃!𝑥 ∈ 𝐴 ∃*𝑦 ∈ 𝐵 𝜑 → ∃*𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑)

Proof of Theorem 2reurmo

Step	Hyp	Ref	Expression
1		reuimrmo 3710	. 2 ⊢ (∀𝑥 ∈ 𝐴 (∃!𝑦 ∈ 𝐵 𝜑 → ∃*𝑦 ∈ 𝐵 𝜑) → (∃!𝑥 ∈ 𝐴 ∃*𝑦 ∈ 𝐵 𝜑 → ∃*𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑))
2		reurmo 3374	. . 3 ⊢ (∃!𝑦 ∈ 𝐵 𝜑 → ∃*𝑦 ∈ 𝐵 𝜑)
3	2	a1i 11	. 2 ⊢ (𝑥 ∈ 𝐴 → (∃!𝑦 ∈ 𝐵 𝜑 → ∃*𝑦 ∈ 𝐵 𝜑))
4	1, 3	mprg 3087	1 ⊢ (∃!𝑥 ∈ 𝐴 ∃*𝑦 ∈ 𝐵 𝜑 → ∃*𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2146  ∃!wreu 3369  ∃*wrmo 3370

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 1943  ax-6 2000  ax-7 2041

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-mo 2569  df-eu 2599  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372

This theorem is used by: (None)