Theorem rgen3 3211

Description: Generalization rule for restricted quantification, with three quantifiers. (Contributed by NM, 12-Jan-2008.)

Hypothesis

Ref	Expression
rgen3.1	⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) → 𝜑)

Assertion

Ref	Expression
rgen3	⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑

Distinct variable groups:   𝑦,𝑧,𝐴   𝑧,𝐵   𝑥,𝑦,𝑧

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

Proof of Theorem rgen3

Step	Hyp	Ref	Expression
1		rgen3.1	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) → 𝜑)
2	1	3expa 1136	. . 3 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 ∈ 𝐶) → 𝜑)
3	2	ralrimiva 3158	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ∀𝑧 ∈ 𝐶 𝜑)
4	3	rgen2 3206	1 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   ∈ wcel 2147  ∀wral 3080

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-ral 3081

This theorem is used by:  poseq  8212  isposi  18547  efmndsgrp  19132  smndex1sgrp  19157  xrge0omnd  21805  addcnlem  25239  addcutslem  28420  zsoring  28852  isgrpoi  31157  lnocoi  31416  0lnfn  32644  lnopcoi  32662  reofld  34008  2zrngasgrp  49616  2zrngmsgrp  49623  2zrngALT  49624