Theorem 2reurmo 3742

Description: Double restricted quantification with restricted existential uniqueness and restricted "at most one", analogous to 2eumo 2727. (Contributed by Alexander van der Vekens, 24-Jun-2017.)

Assertion

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

Distinct variable groups:   𝑦,𝐴   𝑥,𝑦   𝑥,𝐵

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑦)

Proof of Theorem 2reurmo

Step	Hyp	Ref	Expression
1		reuimrmo 3727	. 2 ⊢ (∀𝑥 ∈ 𝐴 (∃!𝑦 ∈ 𝐵 𝜑 → ∃*𝑦 ∈ 𝐵 𝜑) → (∃!𝑥 ∈ 𝐴 ∃*𝑦 ∈ 𝐵 𝜑 → ∃*𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑))
2		reurmo 3425	. . 3 ⊢ (∃!𝑦 ∈ 𝐵 𝜑 → ∃*𝑦 ∈ 𝐵 𝜑)
3	2	a1i 11	. 2 ⊢ (𝑥 ∈ 𝐴 → (∃!𝑦 ∈ 𝐵 𝜑 → ∃*𝑦 ∈ 𝐵 𝜑))
4	1, 3	mprg 3152	1 ⊢ (∃!𝑥 ∈ 𝐴 ∃*𝑦 ∈ 𝐵 𝜑 → ∃*𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑)

Syntax hints:   → wi 4   ∈ wcel 2114  ∃!wreu 3140  ∃*wrmo 3141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-mo 2622  df-eu 2654  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146

This theorem is referenced by: (None)