ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  exlimd2 GIF version

Theorem exlimd2 1557
Description: Deduction from Theorem 19.23 of [Margaris] p. 90. Similar to exlimdh 1558 but with one slightly different hypothesis. (Contributed by Jim Kingdon, 30-Dec-2017.)
Hypotheses
Ref Expression
exlimd2.1 (𝜑 → ∀𝑥𝜑)
exlimd2.2 (𝜑 → (𝜒 → ∀𝑥𝜒))
exlimd2.3 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
exlimd2 (𝜑 → (∃𝑥𝜓𝜒))

Proof of Theorem exlimd2
StepHypRef Expression
1 exlimd2.1 . . 3 (𝜑 → ∀𝑥𝜑)
2 exlimd2.2 . . 3 (𝜑 → (𝜒 → ∀𝑥𝜒))
31, 2alrimih 1428 . 2 (𝜑 → ∀𝑥(𝜒 → ∀𝑥𝜒))
4 exlimd2.3 . . 3 (𝜑 → (𝜓𝜒))
51, 4alrimih 1428 . 2 (𝜑 → ∀𝑥(𝜓𝜒))
6 19.23ht 1456 . . 3 (∀𝑥(𝜒 → ∀𝑥𝜒) → (∀𝑥(𝜓𝜒) ↔ (∃𝑥𝜓𝜒)))
76biimpd 143 . 2 (∀𝑥(𝜒 → ∀𝑥𝜒) → (∀𝑥(𝜓𝜒) → (∃𝑥𝜓𝜒)))
83, 5, 7sylc 62 1 (𝜑 → (∃𝑥𝜓𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1312  wex 1451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-5 1406  ax-gen 1408  ax-ie2 1453
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  equsexd  1690  cbvexdh  1876
  Copyright terms: Public domain W3C validator