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Theorem eeor 1706
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 1694 . . 3 (∃𝑦(𝜑𝜓) ↔ (𝜑 ∨ ∃𝑦𝜓))
32exbii 1616 . 2 (∃𝑥𝑦(𝜑𝜓) ↔ ∃𝑥(𝜑 ∨ ∃𝑦𝜓))
4 eeor.2 . . . 4 𝑥𝜓
54nfex 1648 . . 3 𝑥𝑦𝜓
6519.44 1693 . 2 (∃𝑥(𝜑 ∨ ∃𝑦𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑦𝜓))
73, 6bitri 184 1 (∃𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑦𝜓))
Colors of variables: wff set class
Syntax hints:  wb 105  wo 709  wnf 1471  wex 1503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-ial 1545
This theorem depends on definitions:  df-bi 117  df-nf 1472
This theorem is referenced by: (None)
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