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Theorem bj-stan 13008
 Description: The conjunction of two stable formulas is stable. See bj-stim 13007 for implication, stabnot 818 for negation, and bj-stal 13010 for universal quantification. (Contributed by BJ, 24-Nov-2023.)
Assertion
Ref Expression
bj-stan ((STAB 𝜑STAB 𝜓) → STAB (𝜑𝜓))

Proof of Theorem bj-stan
StepHypRef Expression
1 bj-nnan 13001 . . 3 (¬ ¬ (𝜑𝜓) → (¬ ¬ 𝜑 ∧ ¬ ¬ 𝜓))
2 anim12 341 . . 3 (((¬ ¬ 𝜑𝜑) ∧ (¬ ¬ 𝜓𝜓)) → ((¬ ¬ 𝜑 ∧ ¬ ¬ 𝜓) → (𝜑𝜓)))
31, 2syl5 32 . 2 (((¬ ¬ 𝜑𝜑) ∧ (¬ ¬ 𝜓𝜓)) → (¬ ¬ (𝜑𝜓) → (𝜑𝜓)))
4 df-stab 816 . . 3 (STAB 𝜑 ↔ (¬ ¬ 𝜑𝜑))
5 df-stab 816 . . 3 (STAB 𝜓 ↔ (¬ ¬ 𝜓𝜓))
64, 5anbi12i 455 . 2 ((STAB 𝜑STAB 𝜓) ↔ ((¬ ¬ 𝜑𝜑) ∧ (¬ ¬ 𝜓𝜓)))
7 df-stab 816 . 2 (STAB (𝜑𝜓) ↔ (¬ ¬ (𝜑𝜓) → (𝜑𝜓)))
83, 6, 73imtr4i 200 1 ((STAB 𝜑STAB 𝜓) → STAB (𝜑𝜓))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103  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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