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Theorem 2rmorex 3379
 Description: Double restricted quantification with "at most one," analogous to 2moex 2531. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
Assertion
Ref Expression
2rmorex (∃*𝑥𝐴𝑦𝐵 𝜑 → ∀𝑦𝐵 ∃*𝑥𝐴 𝜑)
Distinct variable groups:   𝑦,𝐴   𝑥,𝐵   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑦)

Proof of Theorem 2rmorex
StepHypRef Expression
1 nfcv 2751 . . 3 𝑦𝐴
2 nfre1 2988 . . 3 𝑦𝑦𝐵 𝜑
31, 2nfrmo 3094 . 2 𝑦∃*𝑥𝐴𝑦𝐵 𝜑
4 rspe 2986 . . . . . 6 ((𝑦𝐵𝜑) → ∃𝑦𝐵 𝜑)
54ex 449 . . . . 5 (𝑦𝐵 → (𝜑 → ∃𝑦𝐵 𝜑))
65ralrimivw 2950 . . . 4 (𝑦𝐵 → ∀𝑥𝐴 (𝜑 → ∃𝑦𝐵 𝜑))
7 rmoim 3374 . . . 4 (∀𝑥𝐴 (𝜑 → ∃𝑦𝐵 𝜑) → (∃*𝑥𝐴𝑦𝐵 𝜑 → ∃*𝑥𝐴 𝜑))
86, 7syl 17 . . 3 (𝑦𝐵 → (∃*𝑥𝐴𝑦𝐵 𝜑 → ∃*𝑥𝐴 𝜑))
98com12 32 . 2 (∃*𝑥𝐴𝑦𝐵 𝜑 → (𝑦𝐵 → ∃*𝑥𝐴 𝜑))
103, 9ralrimi 2940 1 (∃*𝑥𝐴𝑦𝐵 𝜑 → ∀𝑦𝐵 ∃*𝑥𝐴 𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  ∃*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-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 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-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rmo 2904 This theorem is referenced by:  2reu2  39836
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