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Theorem 2rmorex 3608
Description: Double restricted quantification with "at most one", analogous to 2moex 2697. (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 2939 . . 3 𝑦𝐴
2 nfre1 3183 . . 3 𝑦𝑦𝐵 𝜑
31, 2nfrmo 3294 . 2 𝑦∃*𝑥𝐴𝑦𝐵 𝜑
4 rmoim 3603 . . 3 (∀𝑥𝐴 (𝜑 → ∃𝑦𝐵 𝜑) → (∃*𝑥𝐴𝑦𝐵 𝜑 → ∃*𝑥𝐴 𝜑))
5 rspe 3181 . . . . 5 ((𝑦𝐵𝜑) → ∃𝑦𝐵 𝜑)
65ex 402 . . . 4 (𝑦𝐵 → (𝜑 → ∃𝑦𝐵 𝜑))
76ralrimivw 3146 . . 3 (𝑦𝐵 → ∀𝑥𝐴 (𝜑 → ∃𝑦𝐵 𝜑))
84, 7syl11 33 . 2 (∃*𝑥𝐴𝑦𝐵 𝜑 → (𝑦𝐵 → ∃*𝑥𝐴 𝜑))
93, 8ralrimi 3136 1 (∃*𝑥𝐴𝑦𝐵 𝜑 → ∀𝑦𝐵 ∃*𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2157  wral 3087  wrex 3088  ∃*wrmo 3090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-mo 2590  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ral 3092  df-rex 3093  df-rmo 3095
This theorem is referenced by:  2reu2  41952
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