Theorem 3exdistr 1913

Description: Distribution of existential quantifiers. (Contributed by NM, 9-Mar-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Assertion

Ref	Expression
3exdistr	⊢ (∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ∃𝑥(𝜑 ∧ ∃𝑦(𝜓 ∧ ∃𝑧𝜒)))

Distinct variable groups:   𝜑,𝑦   𝜑,𝑧   𝜓,𝑧

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦,𝑧)

Proof of Theorem 3exdistr

Step	Hyp	Ref	Expression
1		3anass 982	. . . 4 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜑 ∧ (𝜓 ∧ 𝜒)))
2	1	2exbii 1604	. . 3 ⊢ (∃𝑦∃𝑧(𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ∃𝑦∃𝑧(𝜑 ∧ (𝜓 ∧ 𝜒)))
3		19.42vv 1909	. . 3 ⊢ (∃𝑦∃𝑧(𝜑 ∧ (𝜓 ∧ 𝜒)) ↔ (𝜑 ∧ ∃𝑦∃𝑧(𝜓 ∧ 𝜒)))
4		exdistr 1907	. . . 4 ⊢ (∃𝑦∃𝑧(𝜓 ∧ 𝜒) ↔ ∃𝑦(𝜓 ∧ ∃𝑧𝜒))
5	4	anbi2i 457	. . 3 ⊢ ((𝜑 ∧ ∃𝑦∃𝑧(𝜓 ∧ 𝜒)) ↔ (𝜑 ∧ ∃𝑦(𝜓 ∧ ∃𝑧𝜒)))
6	2, 3, 5	3bitri 206	. 2 ⊢ (∃𝑦∃𝑧(𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (𝜑 ∧ ∃𝑦(𝜓 ∧ ∃𝑧𝜒)))
7	6	exbii 1603	1 ⊢ (∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ∃𝑥(𝜑 ∧ ∃𝑦(𝜓 ∧ ∃𝑧𝜒)))

Syntax hints:   ∧ wa 104   ↔ wb 105   ∧ w3a 978  ∃wex 1490

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1445  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-4 1508  ax-17 1524  ax-ial 1532

This theorem depends on definitions:  df-bi 117  df-3an 980

This theorem is referenced by: (None)