MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hbe1a Structured version   Visualization version   GIF version

Theorem hbe1a 2132
Description: Dual statement of hbe1 2131. Modified version of axc7e 2303 with a universally quantified consequent. (Contributed by Wolf Lammen, 15-Sep-2021.)
Assertion
Ref Expression
hbe1a (∃𝑥𝑥𝜑 → ∀𝑥𝜑)

Proof of Theorem hbe1a
StepHypRef Expression
1 df-ex 1774 . 2 (∃𝑥𝑥𝜑 ↔ ¬ ∀𝑥 ¬ ∀𝑥𝜑)
2 hbn1 2130 . . 3 (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
32con1i 147 . 2 (¬ ∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥𝜑)
41, 3sylbi 216 1 (∃𝑥𝑥𝜑 → ∀𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1531  wex 1773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-10 2129
This theorem depends on definitions:  df-bi 206  df-ex 1774
This theorem is referenced by:  nf5-1  2133  axc7e  2303  nfeqf2  2368  bj-19.41al  36027  bj-subst  36029  bj-modal4  36083  bj-wnf2  36087  bj-substax12  36090  bj-nnfa1  36128
  Copyright terms: Public domain W3C validator