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Theorem rspec3 3269
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 3268 . . 3 ((𝑥𝐴𝑦𝐵) → ∀𝑧𝐶 𝜑)
32r19.21bi 3240 . 2 (((𝑥𝐴𝑦𝐵) ∧ 𝑧𝐶) → 𝜑)
433impa 1107 1 ((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1084  wcel 2098  wral 3053
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-12 2163
This theorem depends on definitions:  df-bi 206  df-an 396  df-3an 1086  df-ex 1774  df-ral 3054
This theorem is referenced by: (None)
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