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Theorem eeeanv 2342
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 2341 . . 3 (∃𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))
21anbi1i 623 . 2 ((∃𝑥𝑦(𝜑𝜓) ∧ ∃𝑧𝜒) ↔ ((∃𝑥𝜑 ∧ ∃𝑦𝜓) ∧ ∃𝑧𝜒))
3 df-3an 1087 . . . . . 6 ((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ 𝜒))
43exbii 1843 . . . . 5 (∃𝑧(𝜑𝜓𝜒) ↔ ∃𝑧((𝜑𝜓) ∧ 𝜒))
5 19.42v 1950 . . . . 5 (∃𝑧((𝜑𝜓) ∧ 𝜒) ↔ ((𝜑𝜓) ∧ ∃𝑧𝜒))
64, 5bitri 275 . . . 4 (∃𝑧(𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ ∃𝑧𝜒))
762exbii 1844 . . 3 (∃𝑥𝑦𝑧(𝜑𝜓𝜒) ↔ ∃𝑥𝑦((𝜑𝜓) ∧ ∃𝑧𝜒))
8 nfv 1910 . . . . . 6 𝑦𝜒
98nfex 2313 . . . . 5 𝑦𝑧𝜒
10919.41 2224 . . . 4 (∃𝑦((𝜑𝜓) ∧ ∃𝑧𝜒) ↔ (∃𝑦(𝜑𝜓) ∧ ∃𝑧𝜒))
1110exbii 1843 . . 3 (∃𝑥𝑦((𝜑𝜓) ∧ ∃𝑧𝜒) ↔ ∃𝑥(∃𝑦(𝜑𝜓) ∧ ∃𝑧𝜒))
12 nfv 1910 . . . . 5 𝑥𝜒
1312nfex 2313 . . . 4 𝑥𝑧𝜒
141319.41 2224 . . 3 (∃𝑥(∃𝑦(𝜑𝜓) ∧ ∃𝑧𝜒) ↔ (∃𝑥𝑦(𝜑𝜓) ∧ ∃𝑧𝜒))
157, 11, 143bitri 297 . 2 (∃𝑥𝑦𝑧(𝜑𝜓𝜒) ↔ (∃𝑥𝑦(𝜑𝜓) ∧ ∃𝑧𝜒))
16 df-3an 1087 . 2 ((∃𝑥𝜑 ∧ ∃𝑦𝜓 ∧ ∃𝑧𝜒) ↔ ((∃𝑥𝜑 ∧ ∃𝑦𝜓) ∧ ∃𝑧𝜒))
172, 15, 163bitr4i 303 1 (∃𝑥𝑦𝑧(𝜑𝜓𝜒) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓 ∧ ∃𝑧𝜒))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395  w3a 1085  wex 1774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-10 2130  ax-11 2147  ax-12 2167
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-ex 1775  df-nf 1779
This theorem is referenced by:  eloprabgaOLD  7529
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