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Theorem 2rmorex 3040
 Description: Double restricted quantification with "at most one," analogous to 2moex 2275. (Contributed by Alexander van der Vekens, 17-Jun-2017.)
Assertion
Ref Expression
2rmorex (∃*x A y B φy B ∃*x A 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 2489 . . 3 yA
2 nfre1 2670 . . 3 yy B φ
31, 2nfrmo 2786 . 2 y∃*x A y B φ
4 rspe 2675 . . . . . 6 ((y B φ) → y B φ)
54ex 423 . . . . 5 (y B → (φy B φ))
65ralrimivw 2698 . . . 4 (y Bx A (φy B φ))
7 rmoim 3035 . . . 4 (x A (φy B φ) → (∃*x A y B φ∃*x A φ))
86, 7syl 15 . . 3 (y B → (∃*x A y B φ∃*x A φ))
98com12 27 . 2 (∃*x A y B φ → (y B∃*x A φ))
103, 9ralrimi 2695 1 (∃*x A y B φy B ∃*x A φ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1710  ∀wral 2614  ∃wrex 2615  ∃*wrmo 2617 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-rex 2620  df-rmo 2622 This theorem is referenced by: (None)
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