ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  19.35-1 GIF version

Theorem 19.35-1 1556
Description: Forward direction of Theorem 19.35 of [Margaris] p. 90. The converse holds for classical logic but not (for all propositions) in intuitionistic logic (Contributed by Mario Carneiro, 2-Feb-2015.)
Assertion
Ref Expression
19.35-1 (∃𝑥(𝜑𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓))

Proof of Theorem 19.35-1
StepHypRef Expression
1 19.29 1552 . . 3 ((∀𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → ∃𝑥(𝜑 ∧ (𝜑𝜓)))
2 pm3.35 339 . . . 4 ((𝜑 ∧ (𝜑𝜓)) → 𝜓)
32eximi 1532 . . 3 (∃𝑥(𝜑 ∧ (𝜑𝜓)) → ∃𝑥𝜓)
41, 3syl 14 . 2 ((∀𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → ∃𝑥𝜓)
54expcom 114 1 (∃𝑥(𝜑𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wal 1283  wex 1422
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-ial 1468
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  19.35i  1557  19.25  1558  19.36-1  1604  19.37-1  1605  spimt  1666  sbequi  1762
  Copyright terms: Public domain W3C validator