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Theorem ee4anv 1886
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 1627 . . 3 (∃𝑦𝑧𝑤(𝜑𝜓) ↔ ∃𝑧𝑦𝑤(𝜑𝜓))
21exbii 1569 . 2 (∃𝑥𝑦𝑧𝑤(𝜑𝜓) ↔ ∃𝑥𝑧𝑦𝑤(𝜑𝜓))
3 eeanv 1884 . . 3 (∃𝑦𝑤(𝜑𝜓) ↔ (∃𝑦𝜑 ∧ ∃𝑤𝜓))
432exbii 1570 . 2 (∃𝑥𝑧𝑦𝑤(𝜑𝜓) ↔ ∃𝑥𝑧(∃𝑦𝜑 ∧ ∃𝑤𝜓))
5 eeanv 1884 . 2 (∃𝑥𝑧(∃𝑦𝜑 ∧ ∃𝑤𝜓) ↔ (∃𝑥𝑦𝜑 ∧ ∃𝑧𝑤𝜓))
62, 4, 53bitri 205 1 (∃𝑥𝑦𝑧𝑤(𝜑𝜓) ↔ (∃𝑥𝑦𝜑 ∧ ∃𝑧𝑤𝜓))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104  wex 1453
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-4 1472  ax-17 1491  ax-ial 1499
This theorem depends on definitions:  df-bi 116  df-nf 1422
This theorem is referenced by:  ee8anv  1887  cgsex4g  2697  th3qlem1  6499  dmaddpq  7155  dmmulpq  7156  ltdcnq  7173  enq0ref  7209  nqpnq0nq  7229  nqnq0a  7230  nqnq0m  7231  genpdisj  7299  axaddcl  7640  axmulcl  7642
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