Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  19.9h GIF version

Theorem 19.9h 1534
 Description: A wff may be existentially quantified with a variable not free in it. Theorem 19.9 of [Margaris] p. 89. (Contributed by FL, 24-Mar-2007.)
Hypothesis
Ref Expression
19.9h.1 (𝜑 → ∀𝑥𝜑)
Assertion
Ref Expression
19.9h (∃𝑥𝜑𝜑)

Proof of Theorem 19.9h
StepHypRef Expression
1 19.9ht 1532 . . 3 (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑𝜑))
2 19.9h.1 . . 3 (𝜑 → ∀𝑥𝜑)
31, 2mpg 1340 . 2 (∃𝑥𝜑𝜑)
4 19.8a 1482 . 2 (𝜑 → ∃𝑥𝜑)
53, 4impbii 117 1 (∃𝑥𝜑𝜑)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1241  ∃wex 1381 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400 This theorem depends on definitions:  df-bi 110 This theorem is referenced by:  19.9  1535  excomim  1553  exdistrfor  1681  sbcof2  1691  ax11ev  1709  19.9v  1751  exists1  1996
 Copyright terms: Public domain W3C validator