Theorem bj-stand 13629

Description: The conjunction of two stable formulas is stable. Deduction form of bj-stan 13628. Its proof is shorter (when counting all steps, including syntactic steps), so one could prove it first and then bj-stan 13628 from it, the usual way. (Contributed by BJ, 24-Nov-2023.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-stand.1	⊢ (𝜑 → STAB 𝜓)
bj-stand.2	⊢ (𝜑 → STAB 𝜒)

Assertion

Ref	Expression
bj-stand	⊢ (𝜑 → STAB (𝜓 ∧ 𝜒))

Proof of Theorem bj-stand

Step	Hyp	Ref	Expression
1		bj-nnan 13617	. . 3 ⊢ (¬ ¬ (𝜓 ∧ 𝜒) → (¬ ¬ 𝜓 ∧ ¬ ¬ 𝜒))
2		bj-stand.1	. . . . 5 ⊢ (𝜑 → STAB 𝜓)
3		df-stab 821	. . . . 5 ⊢ (STAB 𝜓 ↔ (¬ ¬ 𝜓 → 𝜓))
4	2, 3	sylib 121	. . . 4 ⊢ (𝜑 → (¬ ¬ 𝜓 → 𝜓))
5		bj-stand.2	. . . . 5 ⊢ (𝜑 → STAB 𝜒)
6		df-stab 821	. . . . 5 ⊢ (STAB 𝜒 ↔ (¬ ¬ 𝜒 → 𝜒))
7	5, 6	sylib 121	. . . 4 ⊢ (𝜑 → (¬ ¬ 𝜒 → 𝜒))
8	4, 7	anim12d 333	. . 3 ⊢ (𝜑 → ((¬ ¬ 𝜓 ∧ ¬ ¬ 𝜒) → (𝜓 ∧ 𝜒)))
9	1, 8	syl5 32	. 2 ⊢ (𝜑 → (¬ ¬ (𝜓 ∧ 𝜒) → (𝜓 ∧ 𝜒)))
10		df-stab 821	. 2 ⊢ (STAB (𝜓 ∧ 𝜒) ↔ (¬ ¬ (𝜓 ∧ 𝜒) → (𝜓 ∧ 𝜒)))
11	9, 10	sylibr 133	1 ⊢ (𝜑 → STAB (𝜓 ∧ 𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103  STAB wstab 820

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 604  ax-in2 605

This theorem depends on definitions:  df-bi 116  df-stab 821

This theorem is referenced by: (None)