MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eu4 Structured version   Visualization version   GIF version

Theorem eu4 2639
Description: Uniqueness using implicit substitution. (Contributed by NM, 26-Jul-1995.)
Hypothesis
Ref Expression
eu4.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
eu4 (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∀𝑥𝑦((𝜑𝜓) → 𝑥 = 𝑦)))
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem eu4
StepHypRef Expression
1 df-eu 2582 . 2 (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))
2 eu4.1 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
32mo4 2638 . . 3 (∃*𝑥𝜑 ↔ ∀𝑥𝑦((𝜑𝜓) → 𝑥 = 𝑦))
43anbi2i 616 . 2 ((∃𝑥𝜑 ∧ ∃*𝑥𝜑) ↔ (∃𝑥𝜑 ∧ ∀𝑥𝑦((𝜑𝜓) → 𝑥 = 𝑦)))
51, 4bitri 266 1 (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∀𝑥𝑦((𝜑𝜓) → 𝑥 = 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wal 1650  wex 1874  ∃*wmo 2563  ∃!weu 2581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582
This theorem is referenced by:  eueqOLD  3539  euind  3554  eqeuel  4107  uniintsn  4672  eusv1  5028  omeu  7874  eroveu  8050  climeu  14585  pceu  15844  initoeu2lem2  16944  psgneu  18204  gsumval3eu  18585  frgr3vlem2  27575  3vfriswmgrlem  27578  unirep  33951  rlimdmafv  41949  rlimdmafv2  42030
  Copyright terms: Public domain W3C validator