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Mirrors > Home > ILE Home > Th. List > 3syld | GIF version |
Description: Triple syllogism deduction. (Contributed by Jeff Hankins, 4-Aug-2009.) |
Ref | Expression |
---|---|
3syld.1 | ⊢ (𝜑 → (𝜓 → 𝜒)) |
3syld.2 | ⊢ (𝜑 → (𝜒 → 𝜃)) |
3syld.3 | ⊢ (𝜑 → (𝜃 → 𝜏)) |
Ref | Expression |
---|---|
3syld | ⊢ (𝜑 → (𝜓 → 𝜏)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3syld.1 | . . 3 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
2 | 3syld.2 | . . 3 ⊢ (𝜑 → (𝜒 → 𝜃)) | |
3 | 1, 2 | syld 45 | . 2 ⊢ (𝜑 → (𝜓 → 𝜃)) |
4 | 3syld.3 | . 2 ⊢ (𝜑 → (𝜃 → 𝜏)) | |
5 | 3, 4 | syld 45 | 1 ⊢ (𝜑 → (𝜓 → 𝜏)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 |
This theorem is referenced by: xpfi 6922 fodjumkvlemres 7150 enmkvlem 7152 apreap 8521 msqge0 8550 cju 8894 facavg 10697 mulcn2 11291 coprm 12114 rpexp 12123 cnpnei 13352 ismkvnnlem 14423 |
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