MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hbe1 Structured version   Visualization version   GIF version

Theorem hbe1 2176
Description: The setvar 𝑥 is not free in 𝑥𝜑. Corresponds to the axiom (5) of modal logic (see also modal5 2188). (Contributed by NM, 24-Jan-1993.)
Assertion
Ref Expression
hbe1 (∃𝑥𝜑 → ∀𝑥𝑥𝜑)

Proof of Theorem hbe1
StepHypRef Expression
1 df-ex 1853 . 2 (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
2 hbn1 2175 . 2 (¬ ∀𝑥 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ 𝜑)
31, 2hbxfrbi 1900 1 (∃𝑥𝜑 → ∀𝑥𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1629  wex 1852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-10 2174
This theorem depends on definitions:  df-bi 197  df-ex 1853
This theorem is referenced by:  nfe1  2183  nfe1OLD  2189  equs5eALT  2340  nfeqf2  2452  equs5e  2495  axie1  2745  ac6s6  34312  exlimexi  39255  vk15.4j  39259  vk15.4jVD  39672
  Copyright terms: Public domain W3C validator