Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  2reurex Structured version   Visualization version   GIF version

Theorem 2reurex 40515
Description: Double restricted quantification with existential uniqueness, analogous to 2euex 2543. (Contributed by Alexander van der Vekens, 24-Jun-2017.)
Assertion
Ref Expression
2reurex (∃!𝑥𝐴𝑦𝐵 𝜑 → ∃𝑦𝐵 ∃!𝑥𝐴 𝜑)
Distinct variable groups:   𝑦,𝐴   𝑥,𝑦   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑦)

Proof of Theorem 2reurex
StepHypRef Expression
1 reu5 3152 . 2 (∃!𝑥𝐴𝑦𝐵 𝜑 ↔ (∃𝑥𝐴𝑦𝐵 𝜑 ∧ ∃*𝑥𝐴𝑦𝐵 𝜑))
2 rexcom 3093 . . . 4 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑦𝐵𝑥𝐴 𝜑)
3 nfcv 2761 . . . . . 6 𝑦𝐴
4 nfre1 3001 . . . . . 6 𝑦𝑦𝐵 𝜑
53, 4nfrmo 3109 . . . . 5 𝑦∃*𝑥𝐴𝑦𝐵 𝜑
6 rspe 2999 . . . . . . . . . . 11 ((𝑦𝐵𝜑) → ∃𝑦𝐵 𝜑)
76ex 450 . . . . . . . . . 10 (𝑦𝐵 → (𝜑 → ∃𝑦𝐵 𝜑))
87ralrimivw 2963 . . . . . . . . 9 (𝑦𝐵 → ∀𝑥𝐴 (𝜑 → ∃𝑦𝐵 𝜑))
9 rmoim 3394 . . . . . . . . 9 (∀𝑥𝐴 (𝜑 → ∃𝑦𝐵 𝜑) → (∃*𝑥𝐴𝑦𝐵 𝜑 → ∃*𝑥𝐴 𝜑))
108, 9syl 17 . . . . . . . 8 (𝑦𝐵 → (∃*𝑥𝐴𝑦𝐵 𝜑 → ∃*𝑥𝐴 𝜑))
1110impcom 446 . . . . . . 7 ((∃*𝑥𝐴𝑦𝐵 𝜑𝑦𝐵) → ∃*𝑥𝐴 𝜑)
12 rmo5 3155 . . . . . . 7 (∃*𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 → ∃!𝑥𝐴 𝜑))
1311, 12sylib 208 . . . . . 6 ((∃*𝑥𝐴𝑦𝐵 𝜑𝑦𝐵) → (∃𝑥𝐴 𝜑 → ∃!𝑥𝐴 𝜑))
1413ex 450 . . . . 5 (∃*𝑥𝐴𝑦𝐵 𝜑 → (𝑦𝐵 → (∃𝑥𝐴 𝜑 → ∃!𝑥𝐴 𝜑)))
155, 14reximdai 3008 . . . 4 (∃*𝑥𝐴𝑦𝐵 𝜑 → (∃𝑦𝐵𝑥𝐴 𝜑 → ∃𝑦𝐵 ∃!𝑥𝐴 𝜑))
162, 15syl5bi 232 . . 3 (∃*𝑥𝐴𝑦𝐵 𝜑 → (∃𝑥𝐴𝑦𝐵 𝜑 → ∃𝑦𝐵 ∃!𝑥𝐴 𝜑))
1716impcom 446 . 2 ((∃𝑥𝐴𝑦𝐵 𝜑 ∧ ∃*𝑥𝐴𝑦𝐵 𝜑) → ∃𝑦𝐵 ∃!𝑥𝐴 𝜑)
181, 17sylbi 207 1 (∃!𝑥𝐴𝑦𝐵 𝜑 → ∃𝑦𝐵 ∃!𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1987  wral 2908  wrex 2909  ∃!wreu 2910  ∃*wrmo 2911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916
This theorem is referenced by:  2rexreu  40519
  Copyright terms: Public domain W3C validator