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Theorem 2rmorex 3406
Description: Double restricted quantification with "at most one," analogous to 2moex 2541. (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 2762 . . 3 𝑦𝐴
2 nfre1 3002 . . 3 𝑦𝑦𝐵 𝜑
31, 2nfrmo 3110 . 2 𝑦∃*𝑥𝐴𝑦𝐵 𝜑
4 rmoim 3401 . . 3 (∀𝑥𝐴 (𝜑 → ∃𝑦𝐵 𝜑) → (∃*𝑥𝐴𝑦𝐵 𝜑 → ∃*𝑥𝐴 𝜑))
5 rspe 3000 . . . . 5 ((𝑦𝐵𝜑) → ∃𝑦𝐵 𝜑)
65ex 450 . . . 4 (𝑦𝐵 → (𝜑 → ∃𝑦𝐵 𝜑))
76ralrimivw 2964 . . 3 (𝑦𝐵 → ∀𝑥𝐴 (𝜑 → ∃𝑦𝐵 𝜑))
84, 7syl11 33 . 2 (∃*𝑥𝐴𝑦𝐵 𝜑 → (𝑦𝐵 → ∃*𝑥𝐴 𝜑))
93, 8ralrimi 2954 1 (∃*𝑥𝐴𝑦𝐵 𝜑 → ∀𝑦𝐵 ∃*𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1988  wral 2909  wrex 2910  ∃*wrmo 2912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-eu 2472  df-mo 2473  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rmo 2917
This theorem is referenced by:  2reu2  40950
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