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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-stal | GIF version |
Description: The universal quantification of a stable formula is stable. See bj-stim 13742 for implication, stabnot 828 for negation, and bj-stan 13743 for conjunction. (Contributed by BJ, 24-Nov-2023.) |
Ref | Expression |
---|---|
bj-stal | ⊢ (∀𝑥STAB 𝜑 → STAB ∀𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnal 1642 | . . 3 ⊢ (¬ ¬ ∀𝑥𝜑 → ∀𝑥 ¬ ¬ 𝜑) | |
2 | alim 1450 | . . 3 ⊢ (∀𝑥(¬ ¬ 𝜑 → 𝜑) → (∀𝑥 ¬ ¬ 𝜑 → ∀𝑥𝜑)) | |
3 | 1, 2 | syl5 32 | . 2 ⊢ (∀𝑥(¬ ¬ 𝜑 → 𝜑) → (¬ ¬ ∀𝑥𝜑 → ∀𝑥𝜑)) |
4 | df-stab 826 | . . 3 ⊢ (STAB 𝜑 ↔ (¬ ¬ 𝜑 → 𝜑)) | |
5 | 4 | albii 1463 | . 2 ⊢ (∀𝑥STAB 𝜑 ↔ ∀𝑥(¬ ¬ 𝜑 → 𝜑)) |
6 | df-stab 826 | . 2 ⊢ (STAB ∀𝑥𝜑 ↔ (¬ ¬ ∀𝑥𝜑 → ∀𝑥𝜑)) | |
7 | 3, 5, 6 | 3imtr4i 200 | 1 ⊢ (∀𝑥STAB 𝜑 → STAB ∀𝑥𝜑) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 STAB wstab 825 ∀wal 1346 |
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 ax-5 1440 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-4 1503 ax-17 1519 ax-ial 1527 |
This theorem depends on definitions: df-bi 116 df-stab 826 df-tru 1351 df-fal 1354 df-nf 1454 |
This theorem is referenced by: (None) |
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