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Theorem eeeanv 1851
Description: Rearrange existential quantifiers. (Contributed by NM, 26-Jul-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Assertion
Ref Expression
eeeanv (∃𝑥𝑦𝑧(𝜑𝜓𝜒) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓 ∧ ∃𝑧𝜒))
Distinct variable groups:   𝜑,𝑦   𝜑,𝑧   𝑥,𝑧,𝜓   𝑥,𝑦,𝜒
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝜒(𝑧)

Proof of Theorem eeeanv
StepHypRef Expression
1 df-3an 922 . . 3 ((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ 𝜒))
213exbii 1539 . 2 (∃𝑥𝑦𝑧(𝜑𝜓𝜒) ↔ ∃𝑥𝑦𝑧((𝜑𝜓) ∧ 𝜒))
3 eeanv 1850 . . 3 (∃𝑦𝑧((𝜑𝜓) ∧ 𝜒) ↔ (∃𝑦(𝜑𝜓) ∧ ∃𝑧𝜒))
43exbii 1537 . 2 (∃𝑥𝑦𝑧((𝜑𝜓) ∧ 𝜒) ↔ ∃𝑥(∃𝑦(𝜑𝜓) ∧ ∃𝑧𝜒))
5 eeanv 1850 . . . 4 (∃𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))
65anbi1i 446 . . 3 ((∃𝑥𝑦(𝜑𝜓) ∧ ∃𝑧𝜒) ↔ ((∃𝑥𝜑 ∧ ∃𝑦𝜓) ∧ ∃𝑧𝜒))
7 19.41v 1825 . . 3 (∃𝑥(∃𝑦(𝜑𝜓) ∧ ∃𝑧𝜒) ↔ (∃𝑥𝑦(𝜑𝜓) ∧ ∃𝑧𝜒))
8 df-3an 922 . . 3 ((∃𝑥𝜑 ∧ ∃𝑦𝜓 ∧ ∃𝑧𝜒) ↔ ((∃𝑥𝜑 ∧ ∃𝑦𝜓) ∧ ∃𝑧𝜒))
96, 7, 83bitr4i 210 . 2 (∃𝑥(∃𝑦(𝜑𝜓) ∧ ∃𝑧𝜒) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓 ∧ ∃𝑧𝜒))
102, 4, 93bitri 204 1 (∃𝑥𝑦𝑧(𝜑𝜓𝜒) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓 ∧ ∃𝑧𝜒))
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103  w3a 920  wex 1422
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-3an 922  df-nf 1391
This theorem is referenced by:  vtocl3  2666  spc3egv  2700  spc3gv  2701  eloprabga  5670  prarloc  6965
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