ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  19.40 GIF version

Theorem 19.40 1538
Description: Theorem 19.40 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
19.40 (∃𝑥(𝜑𝜓) → (∃𝑥𝜑 ∧ ∃𝑥𝜓))

Proof of Theorem 19.40
StepHypRef Expression
1 exsimpl 1524 . 2 (∃𝑥(𝜑𝜓) → ∃𝑥𝜑)
2 simpr 107 . . 3 ((𝜑𝜓) → 𝜓)
32eximi 1507 . 2 (∃𝑥(𝜑𝜓) → ∃𝑥𝜓)
41, 3jca 294 1 (∃𝑥(𝜑𝜓) → (∃𝑥𝜑 ∧ ∃𝑥𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wex 1397
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  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-ial 1443
This theorem depends on definitions:  df-bi 114
This theorem is referenced by:  19.40-2  1539  19.41h  1591  19.41  1592  exdistrfor  1697  uniin  3628  copsexg  4009  dmin  4571  imadif  5007  imainlem  5008
  Copyright terms: Public domain W3C validator