Theorem rgen3 3181

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 1118	. . 3 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 ∈ 𝐶) → 𝜑)
3	2	ralrimiva 3128	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ∀𝑧 ∈ 𝐶 𝜑)
4	3	rgen2 3176	1 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   ∈ wcel 2113  ∀wral 3051

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-ral 3052

This theorem is referenced by:  poseq  8100  isposi  18246  efmndsgrp  18811  smndex1sgrp  18833  xrge0omnd  21400  addcnlem  24809  addcutslem  27973  zsoring  28405  isgrpoi  30573  lnocoi  30832  0lnfn  32060  lnopcoi  32078  reofld  33424  2zrngasgrp  48492  2zrngmsgrp  48499  2zrngALT  48500