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Theorem rgen3 2557
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 1198 . . 3 (((𝑥𝐴𝑦𝐵) ∧ 𝑧𝐶) → 𝜑)
32ralrimiva 2543 . 2 ((𝑥𝐴𝑦𝐵) → ∀𝑧𝐶 𝜑)
43rgen2 2556 1 𝑥𝐴𝑦𝐵𝑧𝐶 𝜑
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 973  wcel 2141  wral 2448
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 1440  ax-gen 1442  ax-4 1503  ax-17 1519
This theorem depends on definitions:  df-bi 116  df-3an 975  df-nf 1454  df-ral 2453
This theorem is referenced by:  reg3exmidlemwe  4563  ltsopr  7558  ltsosr  7726  ltso  7997  xrltso  9753  addcncntoplem  13345
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