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Theorem eeor 1626
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 1614 . . 3 (∃𝑦(𝜑𝜓) ↔ (𝜑 ∨ ∃𝑦𝜓))
32exbii 1537 . 2 (∃𝑥𝑦(𝜑𝜓) ↔ ∃𝑥(𝜑 ∨ ∃𝑦𝜓))
4 eeor.2 . . . 4 𝑥𝜓
54nfex 1569 . . 3 𝑥𝑦𝜓
6519.44 1613 . 2 (∃𝑥(𝜑 ∨ ∃𝑦𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑦𝜓))
73, 6bitri 182 1 (∃𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑦𝜓))
Colors of variables: wff set class
Syntax hints:  wb 103  wo 662  wnf 1390  wex 1422
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-nf 1391
This theorem is referenced by: (None)
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