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Theorem bj-stim 13007
 Description: A conjunction with a stable consequent is stable. See stabnot 818 for negation and bj-stan 13008 for conjunction. (Contributed by BJ, 24-Nov-2023.)
Assertion
Ref Expression
bj-stim (STAB 𝜓STAB (𝜑𝜓))

Proof of Theorem bj-stim
StepHypRef Expression
1 bj-nnim 13000 . . 3 (¬ ¬ (𝜑𝜓) → (𝜑 → ¬ ¬ 𝜓))
2 imim2 55 . . 3 ((¬ ¬ 𝜓𝜓) → ((𝜑 → ¬ ¬ 𝜓) → (𝜑𝜓)))
31, 2syl5 32 . 2 ((¬ ¬ 𝜓𝜓) → (¬ ¬ (𝜑𝜓) → (𝜑𝜓)))
4 df-stab 816 . 2 (STAB 𝜓 ↔ (¬ ¬ 𝜓𝜓))
5 df-stab 816 . 2 (STAB (𝜑𝜓) ↔ (¬ ¬ (𝜑𝜓) → (𝜑𝜓)))
63, 4, 53imtr4i 200 1 (STAB 𝜓STAB (𝜑𝜓))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4  STAB wstab 815 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 This theorem depends on definitions:  df-bi 116  df-stab 816 This theorem is referenced by: (None)
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