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Mirrors > Home > MPE Home > Th. List > 3jcad | Structured version Visualization version GIF version |
Description: Deduction conjoining the consequents of three implications. (Contributed by NM, 25-Sep-2005.) |
Ref | Expression |
---|---|
3jcad.1 | ⊢ (𝜑 → (𝜓 → 𝜒)) |
3jcad.2 | ⊢ (𝜑 → (𝜓 → 𝜃)) |
3jcad.3 | ⊢ (𝜑 → (𝜓 → 𝜏)) |
Ref | Expression |
---|---|
3jcad | ⊢ (𝜑 → (𝜓 → (𝜒 ∧ 𝜃 ∧ 𝜏))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3jcad.1 | . . . 4 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
2 | 1 | imp 408 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
3 | 3jcad.2 | . . . 4 ⊢ (𝜑 → (𝜓 → 𝜃)) | |
4 | 3 | imp 408 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜃) |
5 | 3jcad.3 | . . . 4 ⊢ (𝜑 → (𝜓 → 𝜏)) | |
6 | 5 | imp 408 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜏) |
7 | 2, 4, 6 | 3jca 1129 | . 2 ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ∧ 𝜃 ∧ 𝜏)) |
8 | 7 | ex 414 | 1 ⊢ (𝜑 → (𝜓 → (𝜒 ∧ 𝜃 ∧ 𝜏))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 206 df-an 398 df-3an 1090 |
This theorem is referenced by: onfununi 8341 uzm1 12860 ixxssixx 13338 iccid 13369 iccsplit 13462 fzen 13518 lmodprop2d 20534 fbun 23344 hausflim 23485 icoopnst 24455 iocopnst 24456 abelth 25953 usgr2pth 29021 shsvs 30576 cnlnssadj 31333 fnrelpredd 34092 cvmlift2lem10 34303 endofsegid 35057 elicc3 35202 areacirclem1 36576 islvol2aN 38463 alrim3con13v 43294 bgoldbtbndlem4 46476 |
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