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Theorem 4exdistr 1922
Description: Distribution of existential quantifiers in a quadruple conjunction. (Contributed by NM, 9-Mar-1995.) (Proof shortened by Wolf Lammen, 20-Jan-2018.)
Assertion
Ref Expression
4exdistr (∃𝑥𝑦𝑧𝑤((𝜑𝜓) ∧ (𝜒𝜃)) ↔ ∃𝑥(𝜑 ∧ ∃𝑦(𝜓 ∧ ∃𝑧(𝜒 ∧ ∃𝑤𝜃))))
Distinct variable groups:   𝜑,𝑦   𝜑,𝑧   𝜑,𝑤   𝜓,𝑧   𝜓,𝑤   𝜒,𝑤
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦,𝑧)   𝜃(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem 4exdistr
StepHypRef Expression
1 19.42v 1916 . . . . 5 (∃𝑤(𝜒𝜃) ↔ (𝜒 ∧ ∃𝑤𝜃))
21anbi2i 729 . . . 4 (((𝜑𝜓) ∧ ∃𝑤(𝜒𝜃)) ↔ ((𝜑𝜓) ∧ (𝜒 ∧ ∃𝑤𝜃)))
3 19.42v 1916 . . . 4 (∃𝑤((𝜑𝜓) ∧ (𝜒𝜃)) ↔ ((𝜑𝜓) ∧ ∃𝑤(𝜒𝜃)))
4 df-3an 1038 . . . 4 ((𝜑𝜓 ∧ (𝜒 ∧ ∃𝑤𝜃)) ↔ ((𝜑𝜓) ∧ (𝜒 ∧ ∃𝑤𝜃)))
52, 3, 43bitr4i 292 . . 3 (∃𝑤((𝜑𝜓) ∧ (𝜒𝜃)) ↔ (𝜑𝜓 ∧ (𝜒 ∧ ∃𝑤𝜃)))
653exbii 1774 . 2 (∃𝑥𝑦𝑧𝑤((𝜑𝜓) ∧ (𝜒𝜃)) ↔ ∃𝑥𝑦𝑧(𝜑𝜓 ∧ (𝜒 ∧ ∃𝑤𝜃)))
7 3exdistr 1921 . 2 (∃𝑥𝑦𝑧(𝜑𝜓 ∧ (𝜒 ∧ ∃𝑤𝜃)) ↔ ∃𝑥(𝜑 ∧ ∃𝑦(𝜓 ∧ ∃𝑧(𝜒 ∧ ∃𝑤𝜃))))
86, 7bitri 264 1 (∃𝑥𝑦𝑧𝑤((𝜑𝜓) ∧ (𝜒𝜃)) ↔ ∃𝑥(𝜑 ∧ ∃𝑦(𝜓 ∧ ∃𝑧(𝜒 ∧ ∃𝑤𝜃))))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  w3a 1036  wex 1702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886
This theorem depends on definitions:  df-bi 197  df-an 386  df-3an 1038  df-ex 1703
This theorem is referenced by: (None)
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