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Mirrors > Home > MPE Home > Th. List > syl6c | Structured version Visualization version GIF version |
Description: Inference combining syl6 35 with contraction. (Contributed by Alan Sare, 2-May-2011.) |
Ref | Expression |
---|---|
syl6c.1 | ⊢ (𝜑 → (𝜓 → 𝜒)) |
syl6c.2 | ⊢ (𝜑 → (𝜓 → 𝜃)) |
syl6c.3 | ⊢ (𝜒 → (𝜃 → 𝜏)) |
Ref | Expression |
---|---|
syl6c | ⊢ (𝜑 → (𝜓 → 𝜏)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | syl6c.2 | . 2 ⊢ (𝜑 → (𝜓 → 𝜃)) | |
2 | syl6c.1 | . . 3 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
3 | syl6c.3 | . . 3 ⊢ (𝜒 → (𝜃 → 𝜏)) | |
4 | 2, 3 | syl6 35 | . 2 ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜏))) |
5 | 1, 4 | mpdd 43 | 1 ⊢ (𝜑 → (𝜓 → 𝜏)) |
Colors of variables: wff setvar 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: syl6ci 71 syldd 72 impbidd 213 pm5.21ndd 384 jcad 516 a2and 842 zorn2lem6 9912 sqreulem 14711 ontopbas 33889 ontgval 33892 ordtoplem 33896 ordcmp 33908 fvineqsneu 34828 jaodd 39390 ee33 41227 sb5ALT 41231 tratrb 41242 onfrALTlem2 41252 onfrALT 41255 ax6e2ndeq 41265 ee22an 41379 sspwtrALT 41528 sspwtrALT2 41529 trintALT 41587 |
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