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Mirrors > Home > ILE Home > Th. List > 3bitr2d | GIF version |
Description: Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.) |
Ref | Expression |
---|---|
3bitr2d.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
3bitr2d.2 | ⊢ (𝜑 → (𝜃 ↔ 𝜒)) |
3bitr2d.3 | ⊢ (𝜑 → (𝜃 ↔ 𝜏)) |
Ref | Expression |
---|---|
3bitr2d | ⊢ (𝜑 → (𝜓 ↔ 𝜏)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3bitr2d.1 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
2 | 3bitr2d.2 | . . 3 ⊢ (𝜑 → (𝜃 ↔ 𝜒)) | |
3 | 1, 2 | bitr4d 190 | . 2 ⊢ (𝜑 → (𝜓 ↔ 𝜃)) |
4 | 3bitr2d.3 | . 2 ⊢ (𝜑 → (𝜃 ↔ 𝜏)) | |
5 | 3, 4 | bitrd 187 | 1 ⊢ (𝜑 → (𝜓 ↔ 𝜏)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: ceqsralt 2647 frecsuclem 6185 indpi 6962 cauappcvgprlemladdru 7276 prsrlt 7393 lesub2 7996 ltsub2 7998 rec11ap 8238 avglt1 8715 rpnegap 9227 modqmuladdnn0 9836 expap0 10046 2shfti 10326 mulreap 10359 minmax 10722 lemininf 10725 modremain 11268 nn0seqcvgd 11362 divgcdcoprm0 11422 |
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