Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > 3imtr4d | Structured version Visualization version GIF version |
Description: More general version of 3imtr4i 295. Useful for converting conditional definitions in a formula. (Contributed by NM, 26-Oct-1995.) |
Ref | Expression |
---|---|
3imtr4d.1 | ⊢ (𝜑 → (𝜓 → 𝜒)) |
3imtr4d.2 | ⊢ (𝜑 → (𝜃 ↔ 𝜓)) |
3imtr4d.3 | ⊢ (𝜑 → (𝜏 ↔ 𝜒)) |
Ref | Expression |
---|---|
3imtr4d | ⊢ (𝜑 → (𝜃 → 𝜏)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3imtr4d.2 | . 2 ⊢ (𝜑 → (𝜃 ↔ 𝜓)) | |
2 | 3imtr4d.1 | . . 3 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
3 | 3imtr4d.3 | . . 3 ⊢ (𝜑 → (𝜏 ↔ 𝜒)) | |
4 | 2, 3 | sylibrd 262 | . 2 ⊢ (𝜑 → (𝜓 → 𝜏)) |
5 | 1, 4 | sylbid 243 | 1 ⊢ (𝜑 → (𝜃 → 𝜏)) |
Copyright terms: Public domain | W3C validator |