Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  r19.37 GIF version

Theorem r19.37 2587
 Description: Restricted version of one direction of Theorem 19.37 of [Margaris] p. 90. In classical logic the converse would hold if 𝐴 has at least one element, but that is not sufficient in intuitionistic logic. (Contributed by FL, 13-May-2012.) (Revised by Mario Carneiro, 11-Dec-2016.)
Hypothesis
Ref Expression
r19.37.1 𝑥𝜑
Assertion
Ref Expression
r19.37 (∃𝑥𝐴 (𝜑𝜓) → (𝜑 → ∃𝑥𝐴 𝜓))

Proof of Theorem r19.37
StepHypRef Expression
1 r19.37.1 . . 3 𝑥𝜑
2 ax-1 6 . . 3 (𝜑 → (𝑥𝐴𝜑))
31, 2ralrimi 2507 . 2 (𝜑 → ∀𝑥𝐴 𝜑)
4 r19.35-1 2585 . 2 (∃𝑥𝐴 (𝜑𝜓) → (∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜓))
53, 4syl5 32 1 (∃𝑥𝐴 (𝜑𝜓) → (𝜑 → ∃𝑥𝐴 𝜓))
 Colors of variables: wff set class Syntax hints:   → wi 4  Ⅎwnf 1437   ∈ wcel 1481  ∀wral 2417  ∃wrex 2418 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-ial 1515 This theorem depends on definitions:  df-bi 116  df-nf 1438  df-ral 2422  df-rex 2423 This theorem is referenced by:  r19.37av  2588
 Copyright terms: Public domain W3C validator