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Theorem bj-stan 14381
Description: The conjunction of two stable formulas is stable. See bj-stim 14380 for implication, stabnot 833 for negation, and bj-stal 14383 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 14370 . . 3 (¬ ¬ (𝜑𝜓) → (¬ ¬ 𝜑 ∧ ¬ ¬ 𝜓))
2 anim12 344 . . 3 (((¬ ¬ 𝜑𝜑) ∧ (¬ ¬ 𝜓𝜓)) → ((¬ ¬ 𝜑 ∧ ¬ ¬ 𝜓) → (𝜑𝜓)))
31, 2syl5 32 . 2 (((¬ ¬ 𝜑𝜑) ∧ (¬ ¬ 𝜓𝜓)) → (¬ ¬ (𝜑𝜓) → (𝜑𝜓)))
4 df-stab 831 . . 3 (STAB 𝜑 ↔ (¬ ¬ 𝜑𝜑))
5 df-stab 831 . . 3 (STAB 𝜓 ↔ (¬ ¬ 𝜓𝜓))
64, 5anbi12i 460 . 2 ((STAB 𝜑STAB 𝜓) ↔ ((¬ ¬ 𝜑𝜑) ∧ (¬ ¬ 𝜓𝜓)))
7 df-stab 831 . 2 (STAB (𝜑𝜓) ↔ (¬ ¬ (𝜑𝜓) → (𝜑𝜓)))
83, 6, 73imtr4i 201 1 ((STAB 𝜑STAB 𝜓) → STAB (𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  STAB wstab 830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615
This theorem depends on definitions:  df-bi 117  df-stab 831
This theorem is referenced by: (None)
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