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Theorem eeeanv 2219
Description: Rearrange existential quantifiers. (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 2218 . . 3 (∃𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))
21anbi1i 731 . 2 ((∃𝑥𝑦(𝜑𝜓) ∧ ∃𝑧𝜒) ↔ ((∃𝑥𝜑 ∧ ∃𝑦𝜓) ∧ ∃𝑧𝜒))
3 df-3an 1056 . . . . . 6 ((𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ 𝜒))
43exbii 1814 . . . . 5 (∃𝑧(𝜑𝜓𝜒) ↔ ∃𝑧((𝜑𝜓) ∧ 𝜒))
5 19.42v 1921 . . . . 5 (∃𝑧((𝜑𝜓) ∧ 𝜒) ↔ ((𝜑𝜓) ∧ ∃𝑧𝜒))
64, 5bitri 264 . . . 4 (∃𝑧(𝜑𝜓𝜒) ↔ ((𝜑𝜓) ∧ ∃𝑧𝜒))
762exbii 1815 . . 3 (∃𝑥𝑦𝑧(𝜑𝜓𝜒) ↔ ∃𝑥𝑦((𝜑𝜓) ∧ ∃𝑧𝜒))
8 nfv 1883 . . . . . 6 𝑦𝜒
98nfex 2192 . . . . 5 𝑦𝑧𝜒
10919.41 2141 . . . 4 (∃𝑦((𝜑𝜓) ∧ ∃𝑧𝜒) ↔ (∃𝑦(𝜑𝜓) ∧ ∃𝑧𝜒))
1110exbii 1814 . . 3 (∃𝑥𝑦((𝜑𝜓) ∧ ∃𝑧𝜒) ↔ ∃𝑥(∃𝑦(𝜑𝜓) ∧ ∃𝑧𝜒))
12 nfv 1883 . . . . 5 𝑥𝜒
1312nfex 2192 . . . 4 𝑥𝑧𝜒
141319.41 2141 . . 3 (∃𝑥(∃𝑦(𝜑𝜓) ∧ ∃𝑧𝜒) ↔ (∃𝑥𝑦(𝜑𝜓) ∧ ∃𝑧𝜒))
157, 11, 143bitri 286 . 2 (∃𝑥𝑦𝑧(𝜑𝜓𝜒) ↔ (∃𝑥𝑦(𝜑𝜓) ∧ ∃𝑧𝜒))
16 df-3an 1056 . 2 ((∃𝑥𝜑 ∧ ∃𝑦𝜓 ∧ ∃𝑧𝜒) ↔ ((∃𝑥𝜑 ∧ ∃𝑦𝜓) ∧ ∃𝑧𝜒))
172, 15, 163bitr4i 292 1 (∃𝑥𝑦𝑧(𝜑𝜓𝜒) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓 ∧ ∃𝑧𝜒))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  w3a 1054  wex 1744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-11 2074  ax-12 2087
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-ex 1745  df-nf 1750
This theorem is referenced by:  vtocl3  3293  spc3egv  3328  eloprabga  6789
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