Theorem 4exdistr 1963

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	⊢ (∃𝑥∃𝑦∃𝑧∃𝑤((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ∃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

Step	Hyp	Ref	Expression
1		19.42v 1954	. . . . 5 ⊢ (∃𝑤(𝜒 ∧ 𝜃) ↔ (𝜒 ∧ ∃𝑤𝜃))
2	1	anbi2i 625	. . . 4 ⊢ (((𝜑 ∧ 𝜓) ∧ ∃𝑤(𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ ∃𝑤𝜃)))
3		19.42v 1954	. . . 4 ⊢ (∃𝑤((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ((𝜑 ∧ 𝜓) ∧ ∃𝑤(𝜒 ∧ 𝜃)))
4		df-3an 1086	. . . 4 ⊢ ((𝜑 ∧ 𝜓 ∧ (𝜒 ∧ ∃𝑤𝜃)) ↔ ((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ ∃𝑤𝜃)))
5	2, 3, 4	3bitr4i 306	. . 3 ⊢ (∃𝑤((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ (𝜑 ∧ 𝜓 ∧ (𝜒 ∧ ∃𝑤𝜃)))
6	5	3exbii 1851	. 2 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓 ∧ (𝜒 ∧ ∃𝑤𝜃)))
7		3exdistr 1962	. 2 ⊢ (∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓 ∧ (𝜒 ∧ ∃𝑤𝜃)) ↔ ∃𝑥(𝜑 ∧ ∃𝑦(𝜓 ∧ ∃𝑧(𝜒 ∧ ∃𝑤𝜃))))
8	6, 7	bitri 278	1 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤((𝜑 ∧ 𝜓) ∧ (𝜒 ∧ 𝜃)) ↔ ∃𝑥(𝜑 ∧ ∃𝑦(𝜓 ∧ ∃𝑧(𝜒 ∧ ∃𝑤𝜃))))

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

This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1086  df-ex 1782

This theorem is referenced by: (None)