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Theorem ee4anv 1857
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 1599 . . 3 (∃𝑦𝑧𝑤(𝜑𝜓) ↔ ∃𝑧𝑦𝑤(𝜑𝜓))
21exbii 1541 . 2 (∃𝑥𝑦𝑧𝑤(𝜑𝜓) ↔ ∃𝑥𝑧𝑦𝑤(𝜑𝜓))
3 eeanv 1855 . . 3 (∃𝑦𝑤(𝜑𝜓) ↔ (∃𝑦𝜑 ∧ ∃𝑤𝜓))
432exbii 1542 . 2 (∃𝑥𝑧𝑦𝑤(𝜑𝜓) ↔ ∃𝑥𝑧(∃𝑦𝜑 ∧ ∃𝑤𝜓))
5 eeanv 1855 . 2 (∃𝑥𝑧(∃𝑦𝜑 ∧ ∃𝑤𝜓) ↔ (∃𝑥𝑦𝜑 ∧ ∃𝑧𝑤𝜓))
62, 4, 53bitri 204 1 (∃𝑥𝑦𝑧𝑤(𝜑𝜓) ↔ (∃𝑥𝑦𝜑 ∧ ∃𝑧𝑤𝜓))
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103  wex 1426
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-17 1464  ax-ial 1472
This theorem depends on definitions:  df-bi 115  df-nf 1395
This theorem is referenced by:  ee8anv  1858  cgsex4g  2656  th3qlem1  6392  dmaddpq  6936  dmmulpq  6937  ltdcnq  6954  enq0ref  6990  nqpnq0nq  7010  nqnq0a  7011  nqnq0m  7012  genpdisj  7080  axaddcl  7399  axmulcl  7401
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