ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  19.29 GIF version

Theorem 19.29 1600
Description: Theorem 19.29 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.)
Assertion
Ref Expression
19.29 ((∀𝑥𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑𝜓))

Proof of Theorem 19.29
StepHypRef Expression
1 pm3.2 138 . . . 4 (𝜑 → (𝜓 → (𝜑𝜓)))
21alimi 1432 . . 3 (∀𝑥𝜑 → ∀𝑥(𝜓 → (𝜑𝜓)))
3 exim 1579 . . 3 (∀𝑥(𝜓 → (𝜑𝜓)) → (∃𝑥𝜓 → ∃𝑥(𝜑𝜓)))
42, 3syl 14 . 2 (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥(𝜑𝜓)))
54imp 123 1 ((∀𝑥𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wal 1330  wex 1469
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
This theorem is referenced by:  19.29r  1601  19.29x  1603  19.35-1  1604  equs4  1704  equvini  1732  rexxfrd  4388  funimaexglem  5210  bj-inex  13259
  Copyright terms: Public domain W3C validator