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Theorem rspec3 3073
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 3072 . . 3 ((𝑥𝐴𝑦𝐵) → ∀𝑧𝐶 𝜑)
32r19.21bi 3070 . 2 (((𝑥𝐴𝑦𝐵) ∧ 𝑧𝐶) → 𝜑)
433impa 1101 1 ((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072  wcel 2139  wral 3050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-12 2196
This theorem depends on definitions:  df-bi 197  df-an 385  df-3an 1074  df-ex 1854  df-ral 3055
This theorem is referenced by: (None)
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