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Theorem eeeanv 2360
 Description: Distribute three existential quantifiers over a conjunction. (Contributed by NM, 26-Jul-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.) Reduce distinct variable restrictions. (Revised by Wolf Lammen, 20-Jan-2018.)
Assertion
Ref Expression
eeeanv (∃𝑥𝑦𝑧(𝜑𝜓𝜒) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓 ∧ ∃𝑧𝜒))
Distinct variable groups:   𝜑,𝑦   𝜑,𝑧   𝜓,𝑥   𝜓,𝑧   𝜒,𝑥   𝜒,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝜒(𝑧)

Proof of Theorem eeeanv
StepHypRef Expression
1 eeanv 2359 . . 3 (∃𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))
21anbi1i 626 . 2 ((∃𝑥𝑦(𝜑𝜓) ∧ ∃𝑧𝜒) ↔ ((∃𝑥𝜑 ∧ ∃𝑦𝜓) ∧ ∃𝑧𝜒))
3 df-3an 1086 . . . . . 6 ((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ 𝜒))
43exbii 1849 . . . . 5 (∃𝑧(𝜑𝜓𝜒) ↔ ∃𝑧((𝜑𝜓) ∧ 𝜒))
5 19.42v 1954 . . . . 5 (∃𝑧((𝜑𝜓) ∧ 𝜒) ↔ ((𝜑𝜓) ∧ ∃𝑧𝜒))
64, 5bitri 278 . . . 4 (∃𝑧(𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ ∃𝑧𝜒))
762exbii 1850 . . 3 (∃𝑥𝑦𝑧(𝜑𝜓𝜒) ↔ ∃𝑥𝑦((𝜑𝜓) ∧ ∃𝑧𝜒))
8 nfv 1915 . . . . . 6 𝑦𝜒
98nfex 2332 . . . . 5 𝑦𝑧𝜒
10919.41 2235 . . . 4 (∃𝑦((𝜑𝜓) ∧ ∃𝑧𝜒) ↔ (∃𝑦(𝜑𝜓) ∧ ∃𝑧𝜒))
1110exbii 1849 . . 3 (∃𝑥𝑦((𝜑𝜓) ∧ ∃𝑧𝜒) ↔ ∃𝑥(∃𝑦(𝜑𝜓) ∧ ∃𝑧𝜒))
12 nfv 1915 . . . . 5 𝑥𝜒
1312nfex 2332 . . . 4 𝑥𝑧𝜒
141319.41 2235 . . 3 (∃𝑥(∃𝑦(𝜑𝜓) ∧ ∃𝑧𝜒) ↔ (∃𝑥𝑦(𝜑𝜓) ∧ ∃𝑧𝜒))
157, 11, 143bitri 300 . 2 (∃𝑥𝑦𝑧(𝜑𝜓𝜒) ↔ (∃𝑥𝑦(𝜑𝜓) ∧ ∃𝑧𝜒))
16 df-3an 1086 . 2 ((∃𝑥𝜑 ∧ ∃𝑦𝜓 ∧ ∃𝑧𝜒) ↔ ((∃𝑥𝜑 ∧ ∃𝑦𝜓) ∧ ∃𝑧𝜒))
172, 15, 163bitr4i 306 1 (∃𝑥𝑦𝑧(𝜑𝜓𝜒) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓 ∧ ∃𝑧𝜒))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   ∧ w3a 1084  ∃wex 1781 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-11 2158  ax-12 2175 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-nf 1786 This theorem is referenced by:  eloprabga  7240
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