Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bj-stal GIF version

Theorem bj-stal 12987
Description: The universal quantification of stable formula is stable. See bj-stim 12984 for implication, stabnot 818 for negation, and bj-stan 12985 for conjunction. (Contributed by BJ, 24-Nov-2023.)
Assertion
Ref Expression
bj-stal (∀𝑥STAB 𝜑STAB𝑥𝜑)

Proof of Theorem bj-stal
StepHypRef Expression
1 bj-nnal 12979 . . 3 (¬ ¬ ∀𝑥𝜑 → ∀𝑥 ¬ ¬ 𝜑)
2 alim 1433 . . 3 (∀𝑥(¬ ¬ 𝜑𝜑) → (∀𝑥 ¬ ¬ 𝜑 → ∀𝑥𝜑))
31, 2syl5 32 . 2 (∀𝑥(¬ ¬ 𝜑𝜑) → (¬ ¬ ∀𝑥𝜑 → ∀𝑥𝜑))
4 df-stab 816 . . 3 (STAB 𝜑 ↔ (¬ ¬ 𝜑𝜑))
54albii 1446 . 2 (∀𝑥STAB 𝜑 ↔ ∀𝑥(¬ ¬ 𝜑𝜑))
6 df-stab 816 . 2 (STAB𝑥𝜑 ↔ (¬ ¬ ∀𝑥𝜑 → ∀𝑥𝜑))
73, 5, 63imtr4i 200 1 (∀𝑥STAB 𝜑STAB𝑥𝜑)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  STAB wstab 815  wal 1329
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514
This theorem depends on definitions:  df-bi 116  df-stab 816  df-tru 1334  df-fal 1337  df-nf 1437
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator