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Theorem 2rmorex 3541
Description: Double restricted quantification with "at most one," analogous to 2moex 2669. (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 2890 . . 3 𝑦𝐴
2 nfre1 3131 . . 3 𝑦𝑦𝐵 𝜑
31, 2nfrmo 3241 . 2 𝑦∃*𝑥𝐴𝑦𝐵 𝜑
4 rmoim 3536 . . 3 (∀𝑥𝐴 (𝜑 → ∃𝑦𝐵 𝜑) → (∃*𝑥𝐴𝑦𝐵 𝜑 → ∃*𝑥𝐴 𝜑))
5 rspe 3129 . . . . 5 ((𝑦𝐵𝜑) → ∃𝑦𝐵 𝜑)
65ex 449 . . . 4 (𝑦𝐵 → (𝜑 → ∃𝑦𝐵 𝜑))
76ralrimivw 3093 . . 3 (𝑦𝐵 → ∀𝑥𝐴 (𝜑 → ∃𝑦𝐵 𝜑))
84, 7syl11 33 . 2 (∃*𝑥𝐴𝑦𝐵 𝜑 → (𝑦𝐵 → ∃*𝑥𝐴 𝜑))
93, 8ralrimi 3083 1 (∃*𝑥𝐴𝑦𝐵 𝜑 → ∀𝑦𝐵 ∃*𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2127  wral 3038  wrex 3039  ∃*wrmo 3041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1623  df-ex 1842  df-nf 1847  df-eu 2599  df-mo 2600  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ral 3043  df-rex 3044  df-rmo 3046
This theorem is referenced by:  2reu2  41662
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