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Theorem 2rmorex 3760
Description: Double restricted quantification with "at most one", analogous to 2moex 2640. (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 2905 . . 3 𝑦𝐴
2 nfre1 3285 . . 3 𝑦𝑦𝐵 𝜑
31, 2nfrmow 3413 . 2 𝑦∃*𝑥𝐴𝑦𝐵 𝜑
4 rmoim 3746 . . 3 (∀𝑥𝐴 (𝜑 → ∃𝑦𝐵 𝜑) → (∃*𝑥𝐴𝑦𝐵 𝜑 → ∃*𝑥𝐴 𝜑))
5 rspe 3249 . . . . 5 ((𝑦𝐵𝜑) → ∃𝑦𝐵 𝜑)
65ex 412 . . . 4 (𝑦𝐵 → (𝜑 → ∃𝑦𝐵 𝜑))
76ralrimivw 3150 . . 3 (𝑦𝐵 → ∀𝑥𝐴 (𝜑 → ∃𝑦𝐵 𝜑))
84, 7syl11 33 . 2 (∃*𝑥𝐴𝑦𝐵 𝜑 → (𝑦𝐵 → ∃*𝑥𝐴 𝜑))
93, 8ralrimi 3257 1 (∃*𝑥𝐴𝑦𝐵 𝜑 → ∀𝑦𝐵 ∃*𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  wral 3061  wrex 3070  ∃*wrmo 3379
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-10 2141  ax-11 2157  ax-12 2177
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-mo 2540  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rmo 3380
This theorem is referenced by:  2reu2  3898
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