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Theorem impel 484
 Description: An inference for implication elimination. (Contributed by Giovanni Mascellani, 23-May-2019.) (Proof shortened by Wolf Lammen, 2-Sep-2020.)
Hypotheses
Ref Expression
impel.1 (𝜑 → (𝜓𝜒))
impel.2 (𝜃𝜓)
Assertion
Ref Expression
impel ((𝜑𝜃) → 𝜒)

Proof of Theorem impel
StepHypRef Expression
1 impel.2 . . 3 (𝜃𝜓)
2 impel.1 . . 3 (𝜑 → (𝜓𝜒))
31, 2syl5 33 . 2 (𝜑 → (𝜃𝜒))
43imp 444 1 ((𝜑𝜃) → 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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