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Theorem 2reurmo 39831
 Description: Double restricted quantification with restricted existential uniqueness and restricted "at most one.", analogous to 2eumo 2533. (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 39827 . 2 (∀𝑥𝐴 (∃!𝑦𝐵 𝜑 → ∃*𝑦𝐵 𝜑) → (∃!𝑥𝐴 ∃*𝑦𝐵 𝜑 → ∃*𝑥𝐴 ∃!𝑦𝐵 𝜑))
2 reurmo 3138 . . 3 (∃!𝑦𝐵 𝜑 → ∃*𝑦𝐵 𝜑)
32a1i 11 . 2 (𝑥𝐴 → (∃!𝑦𝐵 𝜑 → ∃*𝑦𝐵 𝜑))
41, 3mprg 2910 1 (∃!𝑥𝐴 ∃*𝑦𝐵 𝜑 → ∃*𝑥𝐴 ∃!𝑦𝐵 𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1977  ∃!wreu 2898  ∃*wrmo 2899 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-eu 2462  df-mo 2463  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904 This theorem is referenced by: (None)
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