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

Theorem 2reuswap 3392
 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 2915 . . 3 (∃*𝑦𝐵 𝜑 ↔ ∃*𝑦(𝑦𝐵𝜑))
21ralbii 2974 . 2 (∀𝑥𝐴 ∃*𝑦𝐵 𝜑 ↔ ∀𝑥𝐴 ∃*𝑦(𝑦𝐵𝜑))
3 df-ral 2912 . . . 4 (∀𝑥𝐴 ∃*𝑦(𝑦𝐵𝜑) ↔ ∀𝑥(𝑥𝐴 → ∃*𝑦(𝑦𝐵𝜑)))
4 moanimv 2530 . . . . 5 (∃*𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) ↔ (𝑥𝐴 → ∃*𝑦(𝑦𝐵𝜑)))
54albii 1744 . . . 4 (∀𝑥∃*𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) ↔ ∀𝑥(𝑥𝐴 → ∃*𝑦(𝑦𝐵𝜑)))
63, 5bitr4i 267 . . 3 (∀𝑥𝐴 ∃*𝑦(𝑦𝐵𝜑) ↔ ∀𝑥∃*𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)))
7 2euswap 2547 . . . 4 (∀𝑥∃*𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) → (∃!𝑥𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) → ∃!𝑦𝑥(𝑥𝐴 ∧ (𝑦𝐵𝜑))))
8 df-reu 2914 . . . . 5 (∃!𝑥𝐴𝑦𝐵 𝜑 ↔ ∃!𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 𝜑))
9 r19.42v 3084 . . . . . . . 8 (∃𝑦𝐵 (𝑥𝐴𝜑) ↔ (𝑥𝐴 ∧ ∃𝑦𝐵 𝜑))
10 df-rex 2913 . . . . . . . 8 (∃𝑦𝐵 (𝑥𝐴𝜑) ↔ ∃𝑦(𝑦𝐵 ∧ (𝑥𝐴𝜑)))
119, 10bitr3i 266 . . . . . . 7 ((𝑥𝐴 ∧ ∃𝑦𝐵 𝜑) ↔ ∃𝑦(𝑦𝐵 ∧ (𝑥𝐴𝜑)))
12 an12 837 . . . . . . . 8 ((𝑦𝐵 ∧ (𝑥𝐴𝜑)) ↔ (𝑥𝐴 ∧ (𝑦𝐵𝜑)))
1312exbii 1771 . . . . . . 7 (∃𝑦(𝑦𝐵 ∧ (𝑥𝐴𝜑)) ↔ ∃𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)))
1411, 13bitri 264 . . . . . 6 ((𝑥𝐴 ∧ ∃𝑦𝐵 𝜑) ↔ ∃𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)))
1514eubii 2491 . . . . 5 (∃!𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 𝜑) ↔ ∃!𝑥𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)))
168, 15bitri 264 . . . 4 (∃!𝑥𝐴𝑦𝐵 𝜑 ↔ ∃!𝑥𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)))
17 df-reu 2914 . . . . 5 (∃!𝑦𝐵𝑥𝐴 𝜑 ↔ ∃!𝑦(𝑦𝐵 ∧ ∃𝑥𝐴 𝜑))
18 r19.42v 3084 . . . . . . 7 (∃𝑥𝐴 (𝑦𝐵𝜑) ↔ (𝑦𝐵 ∧ ∃𝑥𝐴 𝜑))
19 df-rex 2913 . . . . . . 7 (∃𝑥𝐴 (𝑦𝐵𝜑) ↔ ∃𝑥(𝑥𝐴 ∧ (𝑦𝐵𝜑)))
2018, 19bitr3i 266 . . . . . 6 ((𝑦𝐵 ∧ ∃𝑥𝐴 𝜑) ↔ ∃𝑥(𝑥𝐴 ∧ (𝑦𝐵𝜑)))
2120eubii 2491 . . . . 5 (∃!𝑦(𝑦𝐵 ∧ ∃𝑥𝐴 𝜑) ↔ ∃!𝑦𝑥(𝑥𝐴 ∧ (𝑦𝐵𝜑)))
2217, 21bitri 264 . . . 4 (∃!𝑦𝐵𝑥𝐴 𝜑 ↔ ∃!𝑦𝑥(𝑥𝐴 ∧ (𝑦𝐵𝜑)))
237, 16, 223imtr4g 285 . . 3 (∀𝑥∃*𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) → (∃!𝑥𝐴𝑦𝐵 𝜑 → ∃!𝑦𝐵𝑥𝐴 𝜑))
246, 23sylbi 207 . 2 (∀𝑥𝐴 ∃*𝑦(𝑦𝐵𝜑) → (∃!𝑥𝐴𝑦𝐵 𝜑 → ∃!𝑦𝐵𝑥𝐴 𝜑))
252, 24sylbi 207 1 (∀𝑥𝐴 ∃*𝑦𝐵 𝜑 → (∃!𝑥𝐴𝑦𝐵 𝜑 → ∃!𝑦𝐵𝑥𝐴 𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384  ∀wal 1478  ∃wex 1701   ∈ wcel 1987  ∃!weu 2469  ∃*wmo 2470  ∀wral 2907  ∃wrex 2908  ∃!wreu 2909  ∃*wrmo 2910 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-10 2016  ax-11 2031  ax-12 2044  ax-13 2245 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-eu 2473  df-mo 2474  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915 This theorem is referenced by:  reuxfr2d  4851  reuxfr3d  29178
 Copyright terms: Public domain W3C validator