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

Theorem ralim 2397
Description: Distribution of restricted quantification over implication. (Contributed by NM, 9-Feb-1997.)
Assertion
Ref Expression
ralim (∀𝑥𝐴 (𝜑𝜓) → (∀𝑥𝐴 𝜑 → ∀𝑥𝐴 𝜓))

Proof of Theorem ralim
StepHypRef Expression
1 df-ral 2328 . . 3 (∀𝑥𝐴 (𝜑𝜓) ↔ ∀𝑥(𝑥𝐴 → (𝜑𝜓)))
2 ax-2 6 . . . 4 ((𝑥𝐴 → (𝜑𝜓)) → ((𝑥𝐴𝜑) → (𝑥𝐴𝜓)))
32al2imi 1363 . . 3 (∀𝑥(𝑥𝐴 → (𝜑𝜓)) → (∀𝑥(𝑥𝐴𝜑) → ∀𝑥(𝑥𝐴𝜓)))
41, 3sylbi 118 . 2 (∀𝑥𝐴 (𝜑𝜓) → (∀𝑥(𝑥𝐴𝜑) → ∀𝑥(𝑥𝐴𝜓)))
5 df-ral 2328 . 2 (∀𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴𝜑))
6 df-ral 2328 . 2 (∀𝑥𝐴 𝜓 ↔ ∀𝑥(𝑥𝐴𝜓))
74, 5, 63imtr4g 198 1 (∀𝑥𝐴 (𝜑𝜓) → (∀𝑥𝐴 𝜑 → ∀𝑥𝐴 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1257  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-5 1352  ax-gen 1354
This theorem depends on definitions:  df-bi 114  df-ral 2328
This theorem is referenced by:  ral2imi  2402  trint  3897  peano2  4346  mpteqb  5289  lbzbi  8648  r19.29uz  9819  alzdvds  10166
  Copyright terms: Public domain W3C validator