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Theorem ee4anv 1934
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 1664 . . 3 (∃𝑦𝑧𝑤(𝜑𝜓) ↔ ∃𝑧𝑦𝑤(𝜑𝜓))
21exbii 1605 . 2 (∃𝑥𝑦𝑧𝑤(𝜑𝜓) ↔ ∃𝑥𝑧𝑦𝑤(𝜑𝜓))
3 eeanv 1932 . . 3 (∃𝑦𝑤(𝜑𝜓) ↔ (∃𝑦𝜑 ∧ ∃𝑤𝜓))
432exbii 1606 . 2 (∃𝑥𝑧𝑦𝑤(𝜑𝜓) ↔ ∃𝑥𝑧(∃𝑦𝜑 ∧ ∃𝑤𝜓))
5 eeanv 1932 . 2 (∃𝑥𝑧(∃𝑦𝜑 ∧ ∃𝑤𝜓) ↔ (∃𝑥𝑦𝜑 ∧ ∃𝑧𝑤𝜓))
62, 4, 53bitri 206 1 (∃𝑥𝑦𝑧𝑤(𝜑𝜓) ↔ (∃𝑥𝑦𝜑 ∧ ∃𝑧𝑤𝜓))
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105  wex 1492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534
This theorem depends on definitions:  df-bi 117  df-nf 1461
This theorem is referenced by:  ee8anv  1935  cgsex4g  2774  th3qlem1  6636  dmaddpq  7377  dmmulpq  7378  ltdcnq  7395  enq0ref  7431  nqpnq0nq  7451  nqnq0a  7452  nqnq0m  7453  genpdisj  7521  axaddcl  7862  axmulcl  7864
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