ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  rspec3 GIF version

Theorem rspec3 2426
Description: Specialization rule for restricted quantification. (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 2425 . . 3 ((𝑥𝐴𝑦𝐵) → ∀𝑧𝐶 𝜑)
32r19.21bi 2424 . 2 (((𝑥𝐴𝑦𝐵) ∧ 𝑧𝐶) → 𝜑)
433impa 1110 1 ((𝑥𝐴𝑦𝐵𝑧𝐶) → 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  w3a 896  wcel 1409  wral 2323
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-4 1416
This theorem depends on definitions:  df-bi 114  df-3an 898  df-ral 2328
This theorem is referenced by:  ordsoexmid  4314
  Copyright terms: Public domain W3C validator