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

Theorem exan 1628
Description: Place a conjunct in the scope of an existential quantifier. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Hypothesis
Ref Expression
exan.1 (∃𝑥𝜑𝜓)
Assertion
Ref Expression
exan 𝑥(𝜑𝜓)

Proof of Theorem exan
StepHypRef Expression
1 hbe1 1429 . . . 4 (∃𝑥𝜑 → ∀𝑥𝑥𝜑)
2119.28h 1499 . . 3 (∀𝑥(∃𝑥𝜑𝜓) ↔ (∃𝑥𝜑 ∧ ∀𝑥𝜓))
3 exan.1 . . 3 (∃𝑥𝜑𝜓)
42, 3mpgbi 1386 . 2 (∃𝑥𝜑 ∧ ∀𝑥𝜓)
5 19.29r 1557 . 2 ((∃𝑥𝜑 ∧ ∀𝑥𝜓) → ∃𝑥(𝜑𝜓))
64, 5ax-mp 7 1 𝑥(𝜑𝜓)
Colors of variables: wff set class
Syntax hints:  wa 102  wal 1287  wex 1426
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 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-ial 1472
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  bm1.3ii  3952  bdbm1.3ii  11428
  Copyright terms: Public domain W3C validator