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Theorem eeor 1641
 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 1629 . . 3 (∃𝑦(𝜑𝜓) ↔ (𝜑 ∨ ∃𝑦𝜓))
32exbii 1552 . 2 (∃𝑥𝑦(𝜑𝜓) ↔ ∃𝑥(𝜑 ∨ ∃𝑦𝜓))
4 eeor.2 . . . 4 𝑥𝜓
54nfex 1584 . . 3 𝑥𝑦𝜓
6519.44 1628 . 2 (∃𝑥(𝜑 ∨ ∃𝑦𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑦𝜓))
73, 6bitri 183 1 (∃𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑦𝜓))
 Colors of variables: wff set class Syntax hints:   ↔ wb 104   ∨ wo 670  Ⅎwnf 1404  ∃wex 1436 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-4 1455  ax-ial 1482 This theorem depends on definitions:  df-bi 116  df-nf 1405 This theorem is referenced by: (None)
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