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

Theorem eeanv 2351
Description: Distribute a pair of existential quantifiers over a conjunction. Combination of 19.41v 1949 and 19.42v 1953. For a version requiring fewer axioms but with additional disjoint variable conditions, see exdistrv 1955. (Contributed by NM, 26-Jul-1995.)
Assertion
Ref Expression
eeanv (∃𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))
Distinct variable groups:   𝜑,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem eeanv
StepHypRef Expression
1 nfv 1914 . 2 𝑦𝜑
2 nfv 1914 . 2 𝑥𝜓
31, 2eean 2350 1 (∃𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wex 1779
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-10 2141  ax-11 2157  ax-12 2177
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1780  df-nf 1784
This theorem is referenced by:  eeeanv  2352  ee4anv  2353  ee4anvOLD  2354  ttrcltr  9756
  Copyright terms: Public domain W3C validator