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Mirrors > Home > MPE Home > Th. List > impel | Structured version Visualization version GIF version |
Description: An inference for implication elimination. (Contributed by Giovanni Mascellani, 23-May-2019.) (Proof shortened by Wolf Lammen, 2-Sep-2020.) |
Ref | Expression |
---|---|
impel.1 | ⊢ (𝜑 → (𝜓 → 𝜒)) |
impel.2 | ⊢ (𝜃 → 𝜓) |
Ref | Expression |
---|---|
impel | ⊢ ((𝜑 ∧ 𝜃) → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | impel.2 | . . 3 ⊢ (𝜃 → 𝜓) | |
2 | impel.1 | . . 3 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
3 | 1, 2 | syl5 33 | . 2 ⊢ (𝜑 → (𝜃 → 𝜒)) |
4 | 3 | imp 444 | 1 ⊢ ((𝜑 ∧ 𝜃) → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 |
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 196 df-an 385 |
This theorem is referenced by: equs5e 2337 elabgt 3316 mob2 3353 reusv2lem2 4795 copsex2t 4883 trssord 5657 trsuc 5727 fiint 8122 eqinf 8273 lcmfunsnlem2lem2 15190 tnggrpr 22269 bj-restpw 32226 setindtr 36609 lighneallem4 40065 proththd 40069 ssn0rex 40311 1wlkv0 40859 1wlkp1lem1 40882 1wlkpwwlkf1ouspgr 41076 wspniunwspnon 41130 wwlksext2clwwlk 41231 trlsegvdeglem1 41388 |
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