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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 445 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
3 | 3jcad.2 | . . . 4 ⊢ (𝜑 → (𝜓 → 𝜃)) | |
4 | 3 | imp 445 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜃) |
5 | 3jcad.3 | . . . 4 ⊢ (𝜑 → (𝜓 → 𝜏)) | |
6 | 5 | imp 445 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜏) |
7 | 2, 4, 6 | 3jca 1242 | . 2 ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ∧ 𝜃 ∧ 𝜏)) |
8 | 7 | ex 450 | 1 ⊢ (𝜑 → (𝜓 → (𝜒 ∧ 𝜃 ∧ 𝜏))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 |
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 197 df-an 386 df-3an 1039 |
This theorem is referenced by: onfununi 7438 uzm1 11718 ixxssixx 12189 iccid 12220 iccsplit 12305 fzen 12358 lmodprop2d 18925 fbun 21644 hausflim 21785 icoopnst 22738 iocopnst 22739 abelth 24195 usgr2pth 26660 shsvs 28182 cnlnssadj 28939 cvmlift2lem10 31294 endofsegid 32192 elicc3 32311 areacirclem1 33500 islvol2aN 34878 alrim3con13v 38743 bgoldbtbndlem4 41696 |
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