Theorem imp42 432

Description: An importation inference. (Contributed by NM, 26-Apr-1994.)

Hypothesis

Ref	Expression
imp4.1	⊢ (𝜑 → (𝜓 → (𝜒 → (𝜃 → 𝜏))))

Assertion

Ref	Expression
imp42	⊢ (((𝜑 ∧ (𝜓 ∧ 𝜒)) ∧ 𝜃) → 𝜏)

Proof of Theorem imp42

Step	Hyp	Ref	Expression
1		imp4.1	. . 3 ⊢ (𝜑 → (𝜓 → (𝜒 → (𝜃 → 𝜏))))
2	1	imp32 424	. 2 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → (𝜃 → 𝜏))
3	2	imp 412	1 ⊢ (((𝜑 ∧ (𝜓 ∧ 𝜒)) ∧ 𝜃) → 𝜏)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-an 402

This theorem is used by:  imp55  448  ltexprlem7  11120  fzdif1  13732  iscatd  17840  isposd  18489  pospropd  18492  mulgghm2  21775  ordtbaslem  23499  txbas  23879  nocvxminlem  28133  frgrncvvdeqlem8  30900  grporcan  31113  chirredlem1  32985  cvxpconn  35986  cvxsconn  35987  rngonegmn1l  38855  prnc  38981  reuopreuprim  48577  uhgrimisgrgriclem  48997