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Theorem eean 1822
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 1594 . . 3 (∃𝑦(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑦𝜓))
32exbii 1512 . 2 (∃𝑥𝑦(𝜑𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦𝜓))
4 eean.2 . . . 4 𝑥𝜓
54nfex 1544 . . 3 𝑥𝑦𝜓
6519.41 1592 . 2 (∃𝑥(𝜑 ∧ ∃𝑦𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))
73, 6bitri 177 1 (∃𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))
Colors of variables: wff set class
Syntax hints:  wa 101  wb 102  wnf 1365  wex 1397
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-nf 1366
This theorem is referenced by:  eeanv  1823  reean  2495
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