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

Theorem ralrimd 2397
 Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 16-Feb-2004.)
Hypotheses
Ref Expression
ralrimd.1 𝑥𝜑
ralrimd.2 𝑥𝜓
ralrimd.3 (𝜑 → (𝜓 → (𝑥𝐴𝜒)))
Assertion
Ref Expression
ralrimd (𝜑 → (𝜓 → ∀𝑥𝐴 𝜒))

Proof of Theorem ralrimd
StepHypRef Expression
1 ralrimd.1 . . 3 𝑥𝜑
2 ralrimd.2 . . 3 𝑥𝜓
3 ralrimd.3 . . 3 (𝜑 → (𝜓 → (𝑥𝐴𝜒)))
41, 2, 3alrimd 1501 . 2 (𝜑 → (𝜓 → ∀𝑥(𝑥𝐴𝜒)))
5 df-ral 2311 . 2 (∀𝑥𝐴 𝜒 ↔ ∀𝑥(𝑥𝐴𝜒))
64, 5syl6ibr 151 1 (𝜑 → (𝜓 → ∀𝑥𝐴 𝜒))
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1241  Ⅎwnf 1349   ∈ wcel 1393  ∀wral 2306 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-4 1400 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-ral 2311 This theorem is referenced by:  ralrimdv  2398  fliftfun  5436  fzrevral  8967
 Copyright terms: Public domain W3C validator