Theorem 3reeanv 3217

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 3176	. . 3 ⊢ (∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ∧ ∃𝑧 ∈ 𝐶 𝜒) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ∧ ∃𝑧 ∈ 𝐶 𝜒))
2		reeanv 3216	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜓))
3	1, 2	bianbi 627	. 2 ⊢ (∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ∧ ∃𝑧 ∈ 𝐶 𝜒) ↔ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜓) ∧ ∃𝑧 ∈ 𝐶 𝜒))
4		df-3an 1088	. . . . 5 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒))
5	4	2rexbii 3116	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 ((𝜑 ∧ 𝜓) ∧ 𝜒))
6		reeanv 3216	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 ((𝜑 ∧ 𝜓) ∧ 𝜒) ↔ (∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ∧ ∃𝑧 ∈ 𝐶 𝜒))
7	5, 6	bitri 275	. . 3 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ∧ ∃𝑧 ∈ 𝐶 𝜒))
8	7	rexbii 3082	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 (𝜑 ∧ 𝜓) ∧ ∃𝑧 ∈ 𝐶 𝜒))
9		df-3an 1088	. 2 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜓 ∧ ∃𝑧 ∈ 𝐶 𝜒) ↔ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜓) ∧ ∃𝑧 ∈ 𝐶 𝜒))
10	3, 8, 9	3bitr4i 303	1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐶 (𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑦 ∈ 𝐵 𝜓 ∧ ∃𝑧 ∈ 𝐶 𝜒))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1086  ∃wrex 3059

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-ex 1779  df-ral 3051  df-rex 3060

This theorem is referenced by:  poxp2  8151  poxp3  8158  imasmnd2  18761  imasgrp2  19047  imasrng  20147  imasring  20300  axeuclid  28927  lshpkrlem6  39057