Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  2eu2 Structured version   Visualization version   GIF version

Theorem 2eu2 2674
 Description: Double existential uniqueness. Usage of this theorem is discouraged because it depends on ax-13 2380. (Contributed by NM, 3-Dec-2001.) (New usage is discouraged.)
Assertion
Ref Expression
2eu2 (∃!𝑦𝑥𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ ∃!𝑥𝑦𝜑))

Proof of Theorem 2eu2
StepHypRef Expression
1 eumo 2598 . . 3 (∃!𝑦𝑥𝜑 → ∃*𝑦𝑥𝜑)
2 2moex 2662 . . 3 (∃*𝑦𝑥𝜑 → ∀𝑥∃*𝑦𝜑)
3 2eu1 2672 . . . 4 (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ (∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑)))
4 simpl 487 . . . 4 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) → ∃!𝑥𝑦𝜑)
53, 4syl6bi 256 . . 3 (∀𝑥∃*𝑦𝜑 → (∃!𝑥∃!𝑦𝜑 → ∃!𝑥𝑦𝜑))
61, 2, 53syl 18 . 2 (∃!𝑦𝑥𝜑 → (∃!𝑥∃!𝑦𝜑 → ∃!𝑥𝑦𝜑))
7 2exeu 2668 . . 3 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) → ∃!𝑥∃!𝑦𝜑)
87expcom 418 . 2 (∃!𝑦𝑥𝜑 → (∃!𝑥𝑦𝜑 → ∃!𝑥∃!𝑦𝜑))
96, 8impbid 215 1 (∃!𝑦𝑥𝜑 → (∃!𝑥∃!𝑦𝜑 ↔ ∃!𝑥𝑦𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1537  ∃wex 1782  ∃*wmo 2556  ∃!weu 2588 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-10 2143  ax-11 2159  ax-12 2176  ax-13 2380 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589 This theorem is referenced by:  2eu8  2681
 Copyright terms: Public domain W3C validator