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Theorem rgen3 3201
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 1110 . . 3 (((𝑥𝐴𝑦𝐵) ∧ 𝑧𝐶) → 𝜑)
32ralrimiva 3179 . 2 ((𝑥𝐴𝑦𝐵) → ∀𝑧𝐶 𝜑)
43rgen2 3200 1 𝑥𝐴𝑦𝐵𝑧𝐶 𝜑
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079  wcel 2105  wral 3135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902
This theorem depends on definitions:  df-bi 208  df-an 397  df-3an 1081  df-ral 3140
This theorem is referenced by:  isposi  17554  addcnlem  23399  isgrpoi  28202  lnocoi  28461  0lnfn  29689  lnopcoi  29707  xrge0omnd  30639  reofld  30840  poseq  32992  efmndsgrp  43983  smndex1sgrp  44008  2zrngasgrp  44139  2zrngmsgrp  44146  2zrngALT  44147
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