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  11166  fzdif1  13779  iscatd  17897  isposd  18546  pospropd  18549  mulgghm2  21837  ordtbaslem  23561  txbas  23941  nocvxminlem  28197  frgrncvvdeqlem8  30964  grporcan  31177  chirredlem1  33049  cvxpconn  36215  cvxsconn  36216  rngonegmn1l  39141  prnc  39267  reuopreuprim  48881  uhgrimisgrgriclem  49301