MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  2eu3 Structured version   Visualization version   GIF version

Theorem 2eu3 2653
Description: Double existential uniqueness. Usage of this theorem is discouraged because it depends on ax-13 2370. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 23-Apr-2023.) (New usage is discouraged.)
Assertion
Ref Expression
2eu3 (∀𝑥𝑦(∃*𝑥𝜑 ∨ ∃*𝑦𝜑) → ((∃!𝑥∃!𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑) ↔ (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))

Proof of Theorem 2eu3
StepHypRef Expression
1 nfmo1 2555 . . . . 5 𝑦∃*𝑦𝜑
2119.31 2227 . . . 4 (∀𝑦(∃*𝑥𝜑 ∨ ∃*𝑦𝜑) ↔ (∀𝑦∃*𝑥𝜑 ∨ ∃*𝑦𝜑))
32albii 1821 . . 3 (∀𝑥𝑦(∃*𝑥𝜑 ∨ ∃*𝑦𝜑) ↔ ∀𝑥(∀𝑦∃*𝑥𝜑 ∨ ∃*𝑦𝜑))
4 nfmo1 2555 . . . . 5 𝑥∃*𝑥𝜑
54nfal 2316 . . . 4 𝑥𝑦∃*𝑥𝜑
6519.32 2226 . . 3 (∀𝑥(∀𝑦∃*𝑥𝜑 ∨ ∃*𝑦𝜑) ↔ (∀𝑦∃*𝑥𝜑 ∨ ∀𝑥∃*𝑦𝜑))
73, 6bitri 274 . 2 (∀𝑥𝑦(∃*𝑥𝜑 ∨ ∃*𝑦𝜑) ↔ (∀𝑦∃*𝑥𝜑 ∨ ∀𝑥∃*𝑦𝜑))
8 2eu1 2650 . . . . . . 7 (∀𝑦∃*𝑥𝜑 → (∃!𝑦∃!𝑥𝜑 ↔ (∃!𝑦𝑥𝜑 ∧ ∃!𝑥𝑦𝜑)))
98biimpd 228 . . . . . 6 (∀𝑦∃*𝑥𝜑 → (∃!𝑦∃!𝑥𝜑 → (∃!𝑦𝑥𝜑 ∧ ∃!𝑥𝑦𝜑)))
10 ancom 461 . . . . . 6 ((∃!𝑦𝑥𝜑 ∧ ∃!𝑥𝑦𝜑) ↔ (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑))
119, 10syl6ib 250 . . . . 5 (∀𝑦∃*𝑥𝜑 → (∃!𝑦∃!𝑥𝜑 → (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))
12 2eu1 2650 . . . . . 6 (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))
1312biimpd 228 . . . . 5 (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 → (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))
1411, 13jaoa 954 . . . 4 ((∀𝑦∃*𝑥𝜑 ∨ ∀𝑥∃*𝑦𝜑) → ((∃!𝑦∃!𝑥𝜑 ∧ ∃!𝑥∃!𝑦𝜑) → (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))
1514ancomsd 466 . . 3 ((∀𝑦∃*𝑥𝜑 ∨ ∀𝑥∃*𝑦𝜑) → ((∃!𝑥∃!𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑) → (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))
16 2exeu 2646 . . . 4 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) → ∃!𝑥∃!𝑦𝜑)
17 2exeu 2646 . . . . 5 ((∃!𝑦𝑥𝜑 ∧ ∃!𝑥𝑦𝜑) → ∃!𝑦∃!𝑥𝜑)
1817ancoms 459 . . . 4 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) → ∃!𝑦∃!𝑥𝜑)
1916, 18jca 512 . . 3 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) → (∃!𝑥∃!𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑))
2015, 19impbid1 224 . 2 ((∀𝑦∃*𝑥𝜑 ∨ ∀𝑥∃*𝑦𝜑) → ((∃!𝑥∃!𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑) ↔ (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))
217, 20sylbi 216 1 (∀𝑥𝑦(∃*𝑥𝜑 ∨ ∃*𝑦𝜑) → ((∃!𝑥∃!𝑦𝜑 ∧ ∃!𝑦∃!𝑥𝜑) ↔ (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wo 845  wal 1539  wex 1781  ∃*wmo 2536  ∃!weu 2566
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 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-11 2154  ax-12 2171  ax-13 2370
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-mo 2538  df-eu 2567
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator