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Mirrors > Home > MPE Home > Th. List > 3imtr4g | Structured version Visualization version GIF version |
Description: More general version of 3imtr4i 295. Useful for converting definitions in a formula. (Contributed by NM, 20-May-1996.) (Proof shortened by Wolf Lammen, 20-Dec-2013.) |
Ref | Expression |
---|---|
3imtr4g.1 | ⊢ (𝜑 → (𝜓 → 𝜒)) |
3imtr4g.2 | ⊢ (𝜃 ↔ 𝜓) |
3imtr4g.3 | ⊢ (𝜏 ↔ 𝜒) |
Ref | Expression |
---|---|
3imtr4g | ⊢ (𝜑 → (𝜃 → 𝜏)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3imtr4g.2 | . . 3 ⊢ (𝜃 ↔ 𝜓) | |
2 | 3imtr4g.1 | . . 3 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
3 | 1, 2 | syl5bi 245 | . 2 ⊢ (𝜑 → (𝜃 → 𝜒)) |
4 | 3imtr4g.3 | . 2 ⊢ (𝜏 ↔ 𝜒) | |
5 | 3, 4 | syl6ibr 255 | 1 ⊢ (𝜑 → (𝜃 → 𝜏)) |
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