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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-stim | GIF version |
Description: A conjunction with a stable consequent is stable. See stabnot 828 for negation , bj-stan 13741 for conjunction , and bj-stal 13743 for universal quantification. (Contributed by BJ, 24-Nov-2023.) |
Ref | Expression |
---|---|
bj-stim | ⊢ (STAB 𝜓 → STAB (𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-nnim 13729 | . . 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 826 | . 2 ⊢ (STAB 𝜓 ↔ (¬ ¬ 𝜓 → 𝜓)) | |
5 | df-stab 826 | . 2 ⊢ (STAB (𝜑 → 𝜓) ↔ (¬ ¬ (𝜑 → 𝜓) → (𝜑 → 𝜓))) | |
6 | 3, 4, 5 | 3imtr4i 200 | 1 ⊢ (STAB 𝜓 → STAB (𝜑 → 𝜓)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 STAB wstab 825 |
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 609 ax-in2 610 |
This theorem depends on definitions: df-bi 116 df-stab 826 |
This theorem is referenced by: (None) |
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