Theorem rgen3 3127

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 1116	. . 3 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 ∈ 𝐶) → 𝜑)
3	2	ralrimiva 3107	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ∀𝑧 ∈ 𝐶 𝜑)
4	3	rgen2 3126	1 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 𝜑

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   ∈ wcel 2108  ∀wral 3063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914

This theorem depends on definitions:  df-bi 206  df-an 396  df-3an 1087  df-ral 3068

This theorem is referenced by:  isposi  17957  efmndsgrp  18440  smndex1sgrp  18462  addcnlem  23933  isgrpoi  28761  lnocoi  29020  0lnfn  30248  lnopcoi  30266  xrge0omnd  31239  reofld  31446  poseq  33729  2zrngasgrp  45386  2zrngmsgrp  45393  2zrngALT  45394