MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rgen3 Structured version   Visualization version   GIF version

Theorem rgen3 3172
Description: Generalization rule for restricted quantification, with three quantifiers. (Contributed by NM, 12-Jan-2008.)
Hypothesis
Ref Expression
rgen3.1 ((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑)
Assertion
Ref Expression
rgen3 𝑥𝐴𝑦𝐵𝑧𝐶 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
StepHypRef Expression
1 rgen3.1 . . . 4 ((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑)
213expa 1115 . . 3 (((𝑥𝐴𝑦𝐵) ∧ 𝑧𝐶) → 𝜑)
32ralrimiva 3152 . 2 ((𝑥𝐴𝑦𝐵) → ∀𝑧𝐶 𝜑)
43rgen2 3171 1 𝑥𝐴𝑦𝐵𝑧𝐶 𝜑
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084  wcel 2112  wral 3109
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-ral 3114
This theorem is referenced by:  isposi  17561  efmndsgrp  18046  smndex1sgrp  18068  addcnlem  23472  isgrpoi  28284  lnocoi  28543  0lnfn  29771  lnopcoi  29789  xrge0omnd  30765  reofld  30967  poseq  33203  2zrngasgrp  44551  2zrngmsgrp  44558  2zrngALT  44559
  Copyright terms: Public domain W3C validator