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Theorem ee4anv 1825
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 1570 . . 3 (∃𝑦𝑧𝑤(𝜑𝜓) ↔ ∃𝑧𝑦𝑤(𝜑𝜓))
21exbii 1512 . 2 (∃𝑥𝑦𝑧𝑤(𝜑𝜓) ↔ ∃𝑥𝑧𝑦𝑤(𝜑𝜓))
3 eeanv 1823 . . 3 (∃𝑦𝑤(𝜑𝜓) ↔ (∃𝑦𝜑 ∧ ∃𝑤𝜓))
432exbii 1513 . 2 (∃𝑥𝑧𝑦𝑤(𝜑𝜓) ↔ ∃𝑥𝑧(∃𝑦𝜑 ∧ ∃𝑤𝜓))
5 eeanv 1823 . 2 (∃𝑥𝑧(∃𝑦𝜑 ∧ ∃𝑤𝜓) ↔ (∃𝑥𝑦𝜑 ∧ ∃𝑧𝑤𝜓))
62, 4, 53bitri 199 1 (∃𝑥𝑦𝑧𝑤(𝜑𝜓) ↔ (∃𝑥𝑦𝜑 ∧ ∃𝑧𝑤𝜓))
Colors of variables: wff set class
Syntax hints:  wa 101  wb 102  wex 1397
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-17 1435  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-nf 1366
This theorem is referenced by:  ee8anv  1826  cgsex4g  2608  th3qlem1  6238  dmaddpq  6534  dmmulpq  6535  ltdcnq  6552  enq0ref  6588  nqpnq0nq  6608  nqnq0a  6609  nqnq0m  6610  genpdisj  6678  axaddcl  6997  axmulcl  6999
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