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Theorem rgen3 3005
 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 1284 . . 3 (((𝑥𝐴𝑦𝐵) ∧ 𝑧𝐶) → 𝜑)
32ralrimiva 2995 . 2 ((𝑥𝐴𝑦𝐵) → ∀𝑧𝐶 𝜑)
43rgen2 3004 1 𝑥𝐴𝑦𝐵𝑧𝐶 𝜑
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1054   ∈ wcel 2030  ∀wral 2941 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879 This theorem depends on definitions:  df-bi 197  df-an 385  df-3an 1056  df-ral 2946 This theorem is referenced by:  isposi  17003  addcnlem  22714  isgrpoi  27480  lnocoi  27740  0lnfn  28972  lnopcoi  28990  xrge0omnd  29839  reofld  29968  poseq  31878  2zrngasgrp  42265  2zrngmsgrp  42272  2zrngALT  42273
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