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Theorem 2reurmo 41702
Description: Double restricted quantification with restricted existential uniqueness and restricted "at most one.", analogous to 2eumo 2694. (Contributed by Alexander van der Vekens, 24-Jun-2017.)
Assertion
Ref Expression
2reurmo (∃!𝑥𝐴 ∃*𝑦𝐵 𝜑 → ∃*𝑥𝐴 ∃!𝑦𝐵 𝜑)
Distinct variable groups:   𝑦,𝐴   𝑥,𝑦   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑦)

Proof of Theorem 2reurmo
StepHypRef Expression
1 reuimrmo 41698 . 2 (∀𝑥𝐴 (∃!𝑦𝐵 𝜑 → ∃*𝑦𝐵 𝜑) → (∃!𝑥𝐴 ∃*𝑦𝐵 𝜑 → ∃*𝑥𝐴 ∃!𝑦𝐵 𝜑))
2 reurmo 3310 . . 3 (∃!𝑦𝐵 𝜑 → ∃*𝑦𝐵 𝜑)
32a1i 11 . 2 (𝑥𝐴 → (∃!𝑦𝐵 𝜑 → ∃*𝑦𝐵 𝜑))
41, 3mprg 3075 1 (∃!𝑥𝐴 ∃*𝑦𝐵 𝜑 → ∃*𝑥𝐴 ∃!𝑦𝐵 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2145  ∃!wreu 3063  ∃*wrmo 3064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-10 2174  ax-12 2203
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-ex 1853  df-nf 1858  df-eu 2622  df-mo 2623  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069
This theorem is referenced by: (None)
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