Theorem eeeanv 1863

Description: Rearrange existential quantifiers. (Contributed by NM, 26-Jul-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Assertion

Ref	Expression
eeeanv	⊢ (∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓 ∧ ∃𝑧𝜒))

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

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

Proof of Theorem eeeanv

Step	Hyp	Ref	Expression
1		df-3an 929	. . 3 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒))
2	1	3exbii 1550	. 2 ⊢ (∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ∃𝑥∃𝑦∃𝑧((𝜑 ∧ 𝜓) ∧ 𝜒))
3		eeanv 1862	. . 3 ⊢ (∃𝑦∃𝑧((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ (∃𝑦(𝜑 ∧ 𝜓) ∧ ∃𝑧𝜒))
4	3	exbii 1548	. 2 ⊢ (∃𝑥∃𝑦∃𝑧((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ ∃𝑥(∃𝑦(𝜑 ∧ 𝜓) ∧ ∃𝑧𝜒))
5		eeanv 1862	. . . 4 ⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓))
6	5	anbi1i 447	. . 3 ⊢ ((∃𝑥∃𝑦(𝜑 ∧ 𝜓) ∧ ∃𝑧𝜒) ↔ ((∃𝑥𝜑 ∧ ∃𝑦𝜓) ∧ ∃𝑧𝜒))
7		19.41v 1837	. . 3 ⊢ (∃𝑥(∃𝑦(𝜑 ∧ 𝜓) ∧ ∃𝑧𝜒) ↔ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) ∧ ∃𝑧𝜒))
8		df-3an 929	. . 3 ⊢ ((∃𝑥𝜑 ∧ ∃𝑦𝜓 ∧ ∃𝑧𝜒) ↔ ((∃𝑥𝜑 ∧ ∃𝑦𝜓) ∧ ∃𝑧𝜒))
9	6, 7, 8	3bitr4i 211	. 2 ⊢ (∃𝑥(∃𝑦(𝜑 ∧ 𝜓) ∧ ∃𝑧𝜒) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓 ∧ ∃𝑧𝜒))
10	2, 4, 9	3bitri 205	1 ⊢ (∃𝑥∃𝑦∃𝑧(𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝜓 ∧ ∃𝑧𝜒))

Syntax hints:   ∧ wa 103   ↔ wb 104   ∧ w3a 927  ∃wex 1433

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-4 1452  ax-17 1471  ax-ial 1479

This theorem depends on definitions:  df-bi 116  df-3an 929  df-nf 1402

This theorem is referenced by:  vtocl3  2689  spc3egv  2724  spc3gv  2725  eloprabga  5773  prarloc  7159