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Mirrors > Home > ILE Home > Th. List > stbid | GIF version |
Description: The equivalent of a stable proposition is stable. (Contributed by Jim Kingdon, 12-Aug-2022.) |
Ref | Expression |
---|---|
stbid.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
stbid | ⊢ (𝜑 → (STAB 𝜓 ↔ STAB 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | stbid.1 | . . . . 5 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
2 | 1 | notbid 657 | . . . 4 ⊢ (𝜑 → (¬ 𝜓 ↔ ¬ 𝜒)) |
3 | 2 | notbid 657 | . . 3 ⊢ (𝜑 → (¬ ¬ 𝜓 ↔ ¬ ¬ 𝜒)) |
4 | 3, 1 | imbi12d 233 | . 2 ⊢ (𝜑 → ((¬ ¬ 𝜓 → 𝜓) ↔ (¬ ¬ 𝜒 → 𝜒))) |
5 | df-stab 821 | . 2 ⊢ (STAB 𝜓 ↔ (¬ ¬ 𝜓 → 𝜓)) | |
6 | df-stab 821 | . 2 ⊢ (STAB 𝜒 ↔ (¬ ¬ 𝜒 → 𝜒)) | |
7 | 4, 5, 6 | 3bitr4g 222 | 1 ⊢ (𝜑 → (STAB 𝜓 ↔ STAB 𝜒)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 104 STAB wstab 820 |
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 604 ax-in2 605 |
This theorem depends on definitions: df-bi 116 df-stab 821 |
This theorem is referenced by: dfss4st 3353 cnstab 8537 exmid1stab 13773 |
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