Theorem bj-stan 13628

Description: The conjunction of two stable formulas is stable. See bj-stim 13627 for implication, stabnot 823 for negation, and bj-stal 13630 for universal quantification. (Contributed by BJ, 24-Nov-2023.)

Assertion

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

Proof of Theorem bj-stan

Step	Hyp	Ref	Expression
1		bj-nnan 13617	. . 3 ⊢ (¬ ¬ (𝜑 ∧ 𝜓) → (¬ ¬ 𝜑 ∧ ¬ ¬ 𝜓))
2		anim12 342	. . 3 ⊢ (((¬ ¬ 𝜑 → 𝜑) ∧ (¬ ¬ 𝜓 → 𝜓)) → ((¬ ¬ 𝜑 ∧ ¬ ¬ 𝜓) → (𝜑 ∧ 𝜓)))
3	1, 2	syl5 32	. 2 ⊢ (((¬ ¬ 𝜑 → 𝜑) ∧ (¬ ¬ 𝜓 → 𝜓)) → (¬ ¬ (𝜑 ∧ 𝜓) → (𝜑 ∧ 𝜓)))
4		df-stab 821	. . 3 ⊢ (STAB 𝜑 ↔ (¬ ¬ 𝜑 → 𝜑))
5		df-stab 821	. . 3 ⊢ (STAB 𝜓 ↔ (¬ ¬ 𝜓 → 𝜓))
6	4, 5	anbi12i 456	. 2 ⊢ ((STAB 𝜑 ∧ STAB 𝜓) ↔ ((¬ ¬ 𝜑 → 𝜑) ∧ (¬ ¬ 𝜓 → 𝜓)))
7		df-stab 821	. 2 ⊢ (STAB (𝜑 ∧ 𝜓) ↔ (¬ ¬ (𝜑 ∧ 𝜓) → (𝜑 ∧ 𝜓)))
8	3, 6, 7	3imtr4i 200	1 ⊢ ((STAB 𝜑 ∧ STAB 𝜓) → STAB (𝜑 ∧ 𝜓))

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)