Theorem 2reurmo 3689

Description: Double restricted quantification with restricted existential uniqueness and restricted "at most one", analogous to 2eumo 2644. (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 3675	. 2 ⊢ (∀𝑥 ∈ 𝐴 (∃!𝑦 ∈ 𝐵 𝜑 → ∃*𝑦 ∈ 𝐵 𝜑) → (∃!𝑥 ∈ 𝐴 ∃*𝑦 ∈ 𝐵 𝜑 → ∃*𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑))
2		reurmo 3354	. . 3 ⊢ (∃!𝑦 ∈ 𝐵 𝜑 → ∃*𝑦 ∈ 𝐵 𝜑)
3	2	a1i 11	. 2 ⊢ (𝑥 ∈ 𝐴 → (∃!𝑦 ∈ 𝐵 𝜑 → ∃*𝑦 ∈ 𝐵 𝜑))
4	1, 3	mprg 3077	1 ⊢ (∃!𝑥 ∈ 𝐴 ∃*𝑦 ∈ 𝐵 𝜑 → ∃*𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑)

Syntax hints:   → wi 4   ∈ wcel 2108  ∃!wreu 3065  ∃*wrmo 3066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784  df-mo 2540  df-eu 2569  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071

This theorem is referenced by: (None)