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Theorem 2rmorex 3742
Description: Double restricted quantification with "at most one", analogous to 2moex 2718. (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 2974 . . 3 𝑦𝐴
2 nfre1 3303 . . 3 𝑦𝑦𝐵 𝜑
31, 2nfrmow 3373 . 2 𝑦∃*𝑥𝐴𝑦𝐵 𝜑
4 rmoim 3728 . . 3 (∀𝑥𝐴 (𝜑 → ∃𝑦𝐵 𝜑) → (∃*𝑥𝐴𝑦𝐵 𝜑 → ∃*𝑥𝐴 𝜑))
5 rspe 3301 . . . . 5 ((𝑦𝐵𝜑) → ∃𝑦𝐵 𝜑)
65ex 413 . . . 4 (𝑦𝐵 → (𝜑 → ∃𝑦𝐵 𝜑))
76ralrimivw 3180 . . 3 (𝑦𝐵 → ∀𝑥𝐴 (𝜑 → ∃𝑦𝐵 𝜑))
84, 7syl11 33 . 2 (∃*𝑥𝐴𝑦𝐵 𝜑 → (𝑦𝐵 → ∃*𝑥𝐴 𝜑))
93, 8ralrimi 3213 1 (∃*𝑥𝐴𝑦𝐵 𝜑 → ∀𝑦𝐵 ∃*𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105  wral 3135  wrex 3136  ∃*wrmo 3138
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-mo 2615  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rmo 3143
This theorem is referenced by:  2reu2  3879
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