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

Theorem hbexOLD 2149
Description: Obsolete proof of hbex 2153 as of 16-Oct-2021. (Contributed by NM, 12-Mar-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
hbexOLD.1 (𝜑 → ∀𝑥𝜑)
Assertion
Ref Expression
hbexOLD (∃𝑦𝜑 → ∀𝑥𝑦𝜑)

Proof of Theorem hbexOLD
StepHypRef Expression
1 df-ex 1702 . 2 (∃𝑦𝜑 ↔ ¬ ∀𝑦 ¬ 𝜑)
2 hbexOLD.1 . . . . 5 (𝜑 → ∀𝑥𝜑)
32hbn 2142 . . . 4 𝜑 → ∀𝑥 ¬ 𝜑)
43hbal 2033 . . 3 (∀𝑦 ¬ 𝜑 → ∀𝑥𝑦 ¬ 𝜑)
54hbn 2142 . 2 (¬ ∀𝑦 ¬ 𝜑 → ∀𝑥 ¬ ∀𝑦 ¬ 𝜑)
61, 5hbxfrbi 1749 1 (∃𝑦𝜑 → ∀𝑥𝑦𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1478  wex 1701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-11 2031  ax-12 2044
This theorem depends on definitions:  df-bi 197  df-or 385  df-ex 1702  df-nf 1707
This theorem is referenced by:  nfexOLD  2152
  Copyright terms: Public domain W3C validator