Theorem 2reurmo 3730

Description: Double restricted quantification with restricted existential uniqueness and restricted "at most one", analogous to 2eumo 2635. (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 3716	. 2 ⊢ (∀𝑥 ∈ 𝐴 (∃!𝑦 ∈ 𝐵 𝜑 → ∃*𝑦 ∈ 𝐵 𝜑) → (∃!𝑥 ∈ 𝐴 ∃*𝑦 ∈ 𝐵 𝜑 → ∃*𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑))
2		reurmo 3357	. . 3 ⊢ (∃!𝑦 ∈ 𝐵 𝜑 → ∃*𝑦 ∈ 𝐵 𝜑)
3	2	a1i 11	. 2 ⊢ (𝑥 ∈ 𝐴 → (∃!𝑦 ∈ 𝐵 𝜑 → ∃*𝑦 ∈ 𝐵 𝜑))
4	1, 3	mprg 3050	1 ⊢ (∃!𝑥 ∈ 𝐴 ∃*𝑦 ∈ 𝐵 𝜑 → ∃*𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑)

Syntax hints:   → wi 4   ∈ wcel 2109  ∃!wreu 3352  ∃*wrmo 3353

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

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-mo 2533  df-eu 2562  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355

This theorem is referenced by: (None)