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Theorem eeor 2345
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 2230 . . 3 (∃𝑦(𝜑𝜓) ↔ (𝜑 ∨ ∃𝑦𝜓))
32exbii 1839 . 2 (∃𝑥𝑦(𝜑𝜓) ↔ ∃𝑥(𝜑 ∨ ∃𝑦𝜓))
4 eeor.2 . . . 4 𝑥𝜓
54nfex 2334 . . 3 𝑥𝑦𝜓
6519.44 2229 . 2 (∃𝑥(𝜑 ∨ ∃𝑦𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑦𝜓))
73, 6bitri 276 1 (∃𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑦𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wo 841  wex 1771  wnf 1775
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-10 2136  ax-11 2151  ax-12 2167
This theorem depends on definitions:  df-bi 208  df-or 842  df-ex 1772  df-nf 1776
This theorem is referenced by: (None)
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