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Theorem rspec3 2919
 Description: Specialization rule for restricted quantification, with three quantifiers. (Contributed by NM, 20-Nov-1994.)
Hypothesis
Ref Expression
rspec3.1 𝑥𝐴𝑦𝐵𝑧𝐶 𝜑
Assertion
Ref Expression
rspec3 ((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑)

Proof of Theorem rspec3
StepHypRef Expression
1 rspec3.1 . . . 4 𝑥𝐴𝑦𝐵𝑧𝐶 𝜑
21rspec2 2918 . . 3 ((𝑥𝐴𝑦𝐵) → ∀𝑧𝐶 𝜑)
32r19.21bi 2916 . 2 (((𝑥𝐴𝑦𝐵) ∧ 𝑧𝐶) → 𝜑)
433impa 1251 1 ((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   ∈ wcel 1977  ∀wral 2896 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-an 385  df-3an 1033  df-ex 1696  df-ral 2901 This theorem is referenced by: (None)
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