Theorem impel 280

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 → (𝜓 → 𝜒))
impel.2	⊢ (𝜃 → 𝜓)

Assertion

Ref	Expression
impel	⊢ ((𝜑 ∧ 𝜃) → 𝜒)

Proof of Theorem impel

Step	Hyp	Ref	Expression
1		impel.2	. . 3 ⊢ (𝜃 → 𝜓)
2		impel.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
3	1, 2	syl5 32	. 2 ⊢ (𝜑 → (𝜃 → 𝜒))
4	3	imp 124	1 ⊢ ((𝜑 ∧ 𝜃) → 𝜒)

Syntax hints:   → wi 4   ∧ wa 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107

This theorem is referenced by:  pm4.55dc  947  mapsnd  6936  fiintim  7204  eqinfti  7324  finomni  7444  frecuzrdgrclt  10801  seq3coll  11239  swrdswrd  11422  swrdccatin1  11442  swrdccatin2  11446  fprodsplitsn  12344  nninfctlemfo  12761  unct  13277  isnzr2  14429  dvcnp2cntop  15690  fsumdvdsmul  15985  perfectlem2  15994  upgrwlkcompim  16483  wlkv0  16490  trlsegvdeglem1  16581