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Mirrors > Home > ILE Home > Th. List > 3imtr3d | GIF version |
Description: More general version of 3imtr3i 198. Useful for converting conditional definitions in a formula. (Contributed by NM, 8-Apr-1996.) |
Ref | Expression |
---|---|
3imtr3d.1 | ⊢ (𝜑 → (𝜓 → 𝜒)) |
3imtr3d.2 | ⊢ (𝜑 → (𝜓 ↔ 𝜃)) |
3imtr3d.3 | ⊢ (𝜑 → (𝜒 ↔ 𝜏)) |
Ref | Expression |
---|---|
3imtr3d | ⊢ (𝜑 → (𝜃 → 𝜏)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3imtr3d.2 | . 2 ⊢ (𝜑 → (𝜓 ↔ 𝜃)) | |
2 | 3imtr3d.1 | . . 3 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
3 | 3imtr3d.3 | . . 3 ⊢ (𝜑 → (𝜒 ↔ 𝜏)) | |
4 | 2, 3 | sylibd 147 | . 2 ⊢ (𝜑 → (𝜓 → 𝜏)) |
5 | 1, 4 | sylbird 168 | 1 ⊢ (𝜑 → (𝜃 → 𝜏)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 |
This theorem depends on definitions: df-bi 115 |
This theorem is referenced by: f1imass 5434 fornex 5762 tposfn2 5904 freccl 6016 eroveu 6220 indpi 6532 axcaucvglemres 7065 caucvgrelemcau 9866 |
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