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Theorem eean 1806
 Description: Rearrange existential quantifiers. (Contributed by NM, 27-Oct-2010.) (Revised by Mario Carneiro, 6-Oct-2016.)
Hypotheses
Ref Expression
eean.1 𝑦𝜑
eean.2 𝑥𝜓
Assertion
Ref Expression
eean (∃𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))

Proof of Theorem eean
StepHypRef Expression
1 eean.1 . . . 4 𝑦𝜑
2119.42 1578 . . 3 (∃𝑦(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑦𝜓))
32exbii 1496 . 2 (∃𝑥𝑦(𝜑𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦𝜓))
4 eean.2 . . . 4 𝑥𝜓
54nfex 1528 . . 3 𝑥𝑦𝜓
6519.41 1576 . 2 (∃𝑥(𝜑 ∧ ∃𝑦𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))
73, 6bitri 173 1 (∃𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98  Ⅎwnf 1349  ∃wex 1381 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427 This theorem depends on definitions:  df-bi 110  df-nf 1350 This theorem is referenced by:  eeanv  1807  reean  2478
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