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Mirrors > Home > ILE Home > Th. List > dcbii | GIF version |
Description: Equivalence property for decidability. Inference form. (Contributed by Jim Kingdon, 28-Mar-2018.) |
Ref | Expression |
---|---|
dcbii.1 | ⊢ (𝜑 ↔ 𝜓) |
Ref | Expression |
---|---|
dcbii | ⊢ (DECID 𝜑 ↔ DECID 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dcbii.1 | . 2 ⊢ (𝜑 ↔ 𝜓) | |
2 | dcbiit 825 | . 2 ⊢ ((𝜑 ↔ 𝜓) → (DECID 𝜑 ↔ DECID 𝜓)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (DECID 𝜑 ↔ DECID 𝜓) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 DECID wdc 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 ax-io 699 |
This theorem depends on definitions: df-bi 116 df-dc 821 |
This theorem is referenced by: dcbi 921 dcned 2333 dfrex2dc 2448 euxfr2dc 2897 exmidexmid 4158 pw1fin 6856 dcfi 6926 exfzdc 10143 fprod1p 11500 prmdc 12011 nninfdclemcl 12221 nninfdclemp1 12223 nninfsellemdc 13624 |
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