Theorem 3reeanv 3321

Description: Rearrange three restricted existential quantifiers. (Contributed by Jeff Madsen, 11-Jun-2010.)

Assertion

Ref	Expression
3reeanv	⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜓 ∧ ∃𝑧 ∈ 𝐶 𝜒))

Distinct variable groups:   𝜑,𝑦,𝑧   𝜓,𝑥,𝑧   𝜒,𝑥,𝑦   𝑦,𝐴   𝑥,𝐵,𝑧   𝑥,𝐶,𝑦

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

Proof of Theorem 3reeanv

Step	Hyp	Ref	Expression
1		r19.41v 3300	. . 3 ⊢ (∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ∧ ∃𝑧 ∈ 𝐶 𝜒) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ∧ ∃𝑧 ∈ 𝐶 𝜒))
2		reeanv 3320	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜓))
3	2	anbi1i 626	. . 3 ⊢ ((∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ∧ ∃𝑧 ∈ 𝐶 𝜒) ↔ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜓) ∧ ∃𝑧 ∈ 𝐶 𝜒))
4	1, 3	bitri 278	. 2 ⊢ (∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ∧ ∃𝑧 ∈ 𝐶 𝜒) ↔ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜓) ∧ ∃𝑧 ∈ 𝐶 𝜒))
5		df-3an 1086	. . . . 5 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒))
6	5	2rexbii 3211	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 ((𝜑 ∧ 𝜓) ∧ 𝜒))
7		reeanv 3320	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 ((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ (∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ∧ ∃𝑧 ∈ 𝐶 𝜒))
8	6, 7	bitri 278	. . 3 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ∧ ∃𝑧 ∈ 𝐶 𝜒))
9	8	rexbii 3210	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ∧ ∃𝑧 ∈ 𝐶 𝜒))
10		df-3an 1086	. 2 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜓 ∧ ∃𝑧 ∈ 𝐶 𝜒) ↔ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜓) ∧ ∃𝑧 ∈ 𝐶 𝜒))
11	4, 9, 10	3bitr4i 306	1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜓 ∧ ∃𝑧 ∈ 𝐶 𝜒))

Syntax hints:   ↔ wb 209   ∧ wa 399   ∧ w3a 1084  ∃wrex 3107

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  df-ral 3111  df-rex 3112

This theorem is referenced by:  imasmnd2  17940  imasgrp2  18206  imasring  19365  axeuclid  26757  lshpkrlem6  36411