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

Theorem 2rexreu 39834
Description: Double restricted existential uniqueness implies double restricted uniqueness quantification, analogous to 2exeu 2537. (Contributed by Alexander van der Vekens, 25-Jun-2017.)
Assertion
Ref Expression
2rexreu ((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑) → ∃!𝑥𝐴 ∃!𝑦𝐵 𝜑)
Distinct variable groups:   𝑦,𝐴   𝑥,𝑦   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑦)

Proof of Theorem 2rexreu
StepHypRef Expression
1 reurmo 3138 . . . 4 (∃!𝑥𝐴𝑦𝐵 𝜑 → ∃*𝑥𝐴𝑦𝐵 𝜑)
2 reurex 3137 . . . . 5 (∃!𝑦𝐵 𝜑 → ∃𝑦𝐵 𝜑)
32rmoimi 39825 . . . 4 (∃*𝑥𝐴𝑦𝐵 𝜑 → ∃*𝑥𝐴 ∃!𝑦𝐵 𝜑)
41, 3syl 17 . . 3 (∃!𝑥𝐴𝑦𝐵 𝜑 → ∃*𝑥𝐴 ∃!𝑦𝐵 𝜑)
5 2reurex 39830 . . 3 (∃!𝑦𝐵𝑥𝐴 𝜑 → ∃𝑥𝐴 ∃!𝑦𝐵 𝜑)
64, 5anim12ci 589 . 2 ((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑) → (∃𝑥𝐴 ∃!𝑦𝐵 𝜑 ∧ ∃*𝑥𝐴 ∃!𝑦𝐵 𝜑))
7 reu5 3136 . 2 (∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 ↔ (∃𝑥𝐴 ∃!𝑦𝐵 𝜑 ∧ ∃*𝑥𝐴 ∃!𝑦𝐵 𝜑))
86, 7sylibr 223 1 ((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑) → ∃!𝑥𝐴 ∃!𝑦𝐵 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wrex 2897  ∃!wreu 2898  ∃*wrmo 2899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904
This theorem is referenced by:  2reu1  39835  2reu2  39836  2reu3  39837
  Copyright terms: Public domain W3C validator