Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  2reuswap Structured version   Visualization version   GIF version

Theorem 2reuswap 3687
 Description: A condition allowing swap of uniqueness and existential quantifiers. (Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by NM, 16-Jun-2017.)
Assertion
Ref Expression
2reuswap (∀𝑥𝐴 ∃*𝑦𝐵 𝜑 → (∃!𝑥𝐴𝑦𝐵 𝜑 → ∃!𝑦𝐵𝑥𝐴 𝜑))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐵(𝑦)

Proof of Theorem 2reuswap
StepHypRef Expression
1 df-rmo 3114 . . 3 (∃*𝑦𝐵 𝜑 ↔ ∃*𝑦(𝑦𝐵𝜑))
21ralbii 3133 . 2 (∀𝑥𝐴 ∃*𝑦𝐵 𝜑 ↔ ∀𝑥𝐴 ∃*𝑦(𝑦𝐵𝜑))
3 df-ral 3111 . . . 4 (∀𝑥𝐴 ∃*𝑦(𝑦𝐵𝜑) ↔ ∀𝑥(𝑥𝐴 → ∃*𝑦(𝑦𝐵𝜑)))
4 moanimv 2681 . . . . 5 (∃*𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) ↔ (𝑥𝐴 → ∃*𝑦(𝑦𝐵𝜑)))
54albii 1821 . . . 4 (∀𝑥∃*𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) ↔ ∀𝑥(𝑥𝐴 → ∃*𝑦(𝑦𝐵𝜑)))
63, 5bitr4i 281 . . 3 (∀𝑥𝐴 ∃*𝑦(𝑦𝐵𝜑) ↔ ∀𝑥∃*𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)))
7 2euswapv 2692 . . . 4 (∀𝑥∃*𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) → (∃!𝑥𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) → ∃!𝑦𝑥(𝑥𝐴 ∧ (𝑦𝐵𝜑))))
8 df-reu 3113 . . . . 5 (∃!𝑥𝐴𝑦𝐵 𝜑 ↔ ∃!𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 𝜑))
9 r19.42v 3304 . . . . . . . 8 (∃𝑦𝐵 (𝑥𝐴𝜑) ↔ (𝑥𝐴 ∧ ∃𝑦𝐵 𝜑))
10 df-rex 3112 . . . . . . . 8 (∃𝑦𝐵 (𝑥𝐴𝜑) ↔ ∃𝑦(𝑦𝐵 ∧ (𝑥𝐴𝜑)))
119, 10bitr3i 280 . . . . . . 7 ((𝑥𝐴 ∧ ∃𝑦𝐵 𝜑) ↔ ∃𝑦(𝑦𝐵 ∧ (𝑥𝐴𝜑)))
12 an12 644 . . . . . . . 8 ((𝑦𝐵 ∧ (𝑥𝐴𝜑)) ↔ (𝑥𝐴 ∧ (𝑦𝐵𝜑)))
1312exbii 1849 . . . . . . 7 (∃𝑦(𝑦𝐵 ∧ (𝑥𝐴𝜑)) ↔ ∃𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)))
1411, 13bitri 278 . . . . . 6 ((𝑥𝐴 ∧ ∃𝑦𝐵 𝜑) ↔ ∃𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)))
1514eubii 2645 . . . . 5 (∃!𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 𝜑) ↔ ∃!𝑥𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)))
168, 15bitri 278 . . . 4 (∃!𝑥𝐴𝑦𝐵 𝜑 ↔ ∃!𝑥𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)))
17 df-reu 3113 . . . . 5 (∃!𝑦𝐵𝑥𝐴 𝜑 ↔ ∃!𝑦(𝑦𝐵 ∧ ∃𝑥𝐴 𝜑))
18 r19.42v 3304 . . . . . . 7 (∃𝑥𝐴 (𝑦𝐵𝜑) ↔ (𝑦𝐵 ∧ ∃𝑥𝐴 𝜑))
19 df-rex 3112 . . . . . . 7 (∃𝑥𝐴 (𝑦𝐵𝜑) ↔ ∃𝑥(𝑥𝐴 ∧ (𝑦𝐵𝜑)))
2018, 19bitr3i 280 . . . . . 6 ((𝑦𝐵 ∧ ∃𝑥𝐴 𝜑) ↔ ∃𝑥(𝑥𝐴 ∧ (𝑦𝐵𝜑)))
2120eubii 2645 . . . . 5 (∃!𝑦(𝑦𝐵 ∧ ∃𝑥𝐴 𝜑) ↔ ∃!𝑦𝑥(𝑥𝐴 ∧ (𝑦𝐵𝜑)))
2217, 21bitri 278 . . . 4 (∃!𝑦𝐵𝑥𝐴 𝜑 ↔ ∃!𝑦𝑥(𝑥𝐴 ∧ (𝑦𝐵𝜑)))
237, 16, 223imtr4g 299 . . 3 (∀𝑥∃*𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) → (∃!𝑥𝐴𝑦𝐵 𝜑 → ∃!𝑦𝐵𝑥𝐴 𝜑))
246, 23sylbi 220 . 2 (∀𝑥𝐴 ∃*𝑦(𝑦𝐵𝜑) → (∃!𝑥𝐴𝑦𝐵 𝜑 → ∃!𝑦𝐵𝑥𝐴 𝜑))
252, 24sylbi 220 1 (∀𝑥𝐴 ∃*𝑦𝐵 𝜑 → (∃!𝑥𝐴𝑦𝐵 𝜑 → ∃!𝑦𝐵𝑥𝐴 𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1536  ∃wex 1781   ∈ wcel 2111  ∃*wmo 2596  ∃!weu 2628  ∀wral 3106  ∃wrex 3107  ∃!wreu 3108  ∃*wrmo 3109 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-11 2158  ax-12 2175 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114 This theorem is referenced by:  reuxfrd  3689
 Copyright terms: Public domain W3C validator