Theorem bj-stim 13627

Description: A conjunction with a stable consequent is stable. See stabnot 823 for negation , bj-stan 13628 for conjunction , and bj-stal 13630 for universal quantification. (Contributed by BJ, 24-Nov-2023.)

Assertion

Ref	Expression
bj-stim	⊢ (STAB 𝜓 → STAB (𝜑 → 𝜓))

Proof of Theorem bj-stim

Step	Hyp	Ref	Expression
1		bj-nnim 13616	. . 3 ⊢ (¬ ¬ (𝜑 → 𝜓) → (𝜑 → ¬ ¬ 𝜓))
2		imim2 55	. . 3 ⊢ ((¬ ¬ 𝜓 → 𝜓) → ((𝜑 → ¬ ¬ 𝜓) → (𝜑 → 𝜓)))
3	1, 2	syl5 32	. 2 ⊢ ((¬ ¬ 𝜓 → 𝜓) → (¬ ¬ (𝜑 → 𝜓) → (𝜑 → 𝜓)))
4		df-stab 821	. 2 ⊢ (STAB 𝜓 ↔ (¬ ¬ 𝜓 → 𝜓))
5		df-stab 821	. 2 ⊢ (STAB (𝜑 → 𝜓) ↔ (¬ ¬ (𝜑 → 𝜓) → (𝜑 → 𝜓)))
6	3, 4, 5	3imtr4i 200	1 ⊢ (STAB 𝜓 → STAB (𝜑 → 𝜓))

Syntax hints:  ¬ wn 3   → wi 4  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)