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Theorem ee4anv 2220
Description: Rearrange existential quantifiers. (Contributed by NM, 31-Jul-1995.)
Assertion
Ref Expression
ee4anv (∃𝑥𝑦𝑧𝑤(𝜑𝜓) ↔ (∃𝑥𝑦𝜑 ∧ ∃𝑧𝑤𝜓))
Distinct variable groups:   𝜑,𝑧   𝜑,𝑤   𝜓,𝑥   𝜓,𝑦   𝑦,𝑧   𝑥,𝑤
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑧,𝑤)

Proof of Theorem ee4anv
StepHypRef Expression
1 excom 2082 . . 3 (∃𝑦𝑧𝑤(𝜑𝜓) ↔ ∃𝑧𝑦𝑤(𝜑𝜓))
21exbii 1814 . 2 (∃𝑥𝑦𝑧𝑤(𝜑𝜓) ↔ ∃𝑥𝑧𝑦𝑤(𝜑𝜓))
3 eeanv 2218 . . 3 (∃𝑦𝑤(𝜑𝜓) ↔ (∃𝑦𝜑 ∧ ∃𝑤𝜓))
432exbii 1815 . 2 (∃𝑥𝑧𝑦𝑤(𝜑𝜓) ↔ ∃𝑥𝑧(∃𝑦𝜑 ∧ ∃𝑤𝜓))
5 eeanv 2218 . 2 (∃𝑥𝑧(∃𝑦𝜑 ∧ ∃𝑤𝜓) ↔ (∃𝑥𝑦𝜑 ∧ ∃𝑧𝑤𝜓))
62, 4, 53bitri 286 1 (∃𝑥𝑦𝑧𝑤(𝜑𝜓) ↔ (∃𝑥𝑦𝜑 ∧ ∃𝑧𝑤𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  wex 1744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-11 2074  ax-12 2087
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1745  df-nf 1750
This theorem is referenced by:  cgsex4g  3271  5oalem7  28647  3oalem3  28651  elfuns  32147
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