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Theorem eeeanv 1906
 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 965 . . 3 ((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ 𝜒))
213exbii 1587 . 2 (∃𝑥𝑦𝑧(𝜑𝜓𝜒) ↔ ∃𝑥𝑦𝑧((𝜑𝜓) ∧ 𝜒))
3 eeanv 1905 . . 3 (∃𝑦𝑧((𝜑𝜓) ∧ 𝜒) ↔ (∃𝑦(𝜑𝜓) ∧ ∃𝑧𝜒))
43exbii 1585 . 2 (∃𝑥𝑦𝑧((𝜑𝜓) ∧ 𝜒) ↔ ∃𝑥(∃𝑦(𝜑𝜓) ∧ ∃𝑧𝜒))
5 eeanv 1905 . . . 4 (∃𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))
65anbi1i 454 . . 3 ((∃𝑥𝑦(𝜑𝜓) ∧ ∃𝑧𝜒) ↔ ((∃𝑥𝜑 ∧ ∃𝑦𝜓) ∧ ∃𝑧𝜒))
7 19.41v 1875 . . 3 (∃𝑥(∃𝑦(𝜑𝜓) ∧ ∃𝑧𝜒) ↔ (∃𝑥𝑦(𝜑𝜓) ∧ ∃𝑧𝜒))
8 df-3an 965 . . 3 ((∃𝑥𝜑 ∧ ∃𝑦𝜓 ∧ ∃𝑧𝜒) ↔ ((∃𝑥𝜑 ∧ ∃𝑦𝜓) ∧ ∃𝑧𝜒))
96, 7, 83bitr4i 211 . 2 (∃𝑥(∃𝑦(𝜑𝜓) ∧ ∃𝑧𝜒) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓 ∧ ∃𝑧𝜒))
102, 4, 93bitri 205 1 (∃𝑥𝑦𝑧(𝜑𝜓𝜒) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓 ∧ ∃𝑧𝜒))
 Colors of variables: wff set class Syntax hints:   ∧ wa 103   ↔ wb 104   ∧ w3a 963  ∃wex 1469 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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-17 1507  ax-ial 1515 This theorem depends on definitions:  df-bi 116  df-3an 965  df-nf 1438 This theorem is referenced by:  vtocl3  2743  spc3egv  2778  spc3gv  2779  eloprabga  5862  prarloc  7331
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