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

Theorem eeor 2343
 Description: Rearrange existential quantifiers. (Contributed by NM, 8-Aug-1994.)
Hypotheses
Ref Expression
eeor.1 𝑦𝜑
eeor.2 𝑥𝜓
Assertion
Ref Expression
eeor (∃𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑦𝜓))

Proof of Theorem eeor
StepHypRef Expression
1 eeor.1 . . . 4 𝑦𝜑
2119.45 2238 . . 3 (∃𝑦(𝜑𝜓) ↔ (𝜑 ∨ ∃𝑦𝜓))
32exbii 1849 . 2 (∃𝑥𝑦(𝜑𝜓) ↔ ∃𝑥(𝜑 ∨ ∃𝑦𝜓))
4 eeor.2 . . . 4 𝑥𝜓
54nfex 2332 . . 3 𝑥𝑦𝜓
6519.44 2237 . 2 (∃𝑥(𝜑 ∨ ∃𝑦𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑦𝜓))
73, 6bitri 278 1 (∃𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑦𝜓))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∨ wo 844  ∃wex 1781  Ⅎwnf 1785 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 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-11 2158  ax-12 2175 This theorem depends on definitions:  df-bi 210  df-or 845  df-ex 1782  df-nf 1786 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator