Theorem bj-stal 13784

Description: The universal quantification of a stable formula is stable. See bj-stim 13781 for implication, stabnot 828 for negation, and bj-stan 13782 for conjunction. (Contributed by BJ, 24-Nov-2023.)

Assertion

Ref	Expression
bj-stal	⊢ (∀𝑥STAB 𝜑 → STAB ∀𝑥𝜑)

Proof of Theorem bj-stal

Step	Hyp	Ref	Expression
1		nnal 1642	. . 3 ⊢ (¬ ¬ ∀𝑥𝜑 → ∀𝑥 ¬ ¬ 𝜑)
2		alim 1450	. . 3 ⊢ (∀𝑥(¬ ¬ 𝜑 → 𝜑) → (∀𝑥 ¬ ¬ 𝜑 → ∀𝑥𝜑))
3	1, 2	syl5 32	. 2 ⊢ (∀𝑥(¬ ¬ 𝜑 → 𝜑) → (¬ ¬ ∀𝑥𝜑 → ∀𝑥𝜑))
4		df-stab 826	. . 3 ⊢ (STAB 𝜑 ↔ (¬ ¬ 𝜑 → 𝜑))
5	4	albii 1463	. 2 ⊢ (∀𝑥STAB 𝜑 ↔ ∀𝑥(¬ ¬ 𝜑 → 𝜑))
6		df-stab 826	. 2 ⊢ (STAB ∀𝑥𝜑 ↔ (¬ ¬ ∀𝑥𝜑 → ∀𝑥𝜑))
7	3, 5, 6	3imtr4i 200	1 ⊢ (∀𝑥STAB 𝜑 → STAB ∀𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4  STAB wstab 825  ∀wal 1346

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 609  ax-in2 610  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-stab 826  df-tru 1351  df-fal 1354  df-nf 1454

This theorem is referenced by: (None)