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Theorem rgen3 2519
 Description: Generalization rule for restricted quantification. (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 1181 . . 3 (((𝑥𝐴𝑦𝐵) ∧ 𝑧𝐶) → 𝜑)
32ralrimiva 2505 . 2 ((𝑥𝐴𝑦𝐵) → ∀𝑧𝐶 𝜑)
43rgen2 2518 1 𝑥𝐴𝑦𝐵𝑧𝐶 𝜑
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 962   ∈ wcel 1480  ∀wral 2416 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-4 1487  ax-17 1506 This theorem depends on definitions:  df-bi 116  df-3an 964  df-nf 1437  df-ral 2421 This theorem is referenced by:  reg3exmidlemwe  4493  ltsopr  7411  ltsosr  7579  ltso  7849  xrltso  9589  addcncntoplem  12729
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