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  940  fiintim  7001  eqinfti  7095  finomni  7215  frecuzrdgrclt  10524  seq3coll  10951  fprodsplitsn  11815  nninfctlemfo  12232  unct  12684  isnzr2  13816  dvcnp2cntop  15019  fsumdvdsmul  15311  perfectlem2  15320