Theorem imp42 426

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 418	. 2 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → (𝜃 → 𝜏))
3	2	imp 406	1 ⊢ (((𝜑 ∧ (𝜓 ∧ 𝜒)) ∧ 𝜃) → 𝜏)

Syntax hints:   → wi 4   ∧ wa 395

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 207  df-an 396

This theorem is referenced by:  imp55  442  ltexprlem7  10965  fzdif1  13559  iscatd  17639  isposd  18288  pospropd  18291  mulgghm2  21456  ordtbaslem  23153  txbas  23532  nocvxminlem  27746  frgrncvvdeqlem8  30376  grporcan  30589  chirredlem1  32461  cvxpconn  35424  cvxsconn  35425  rngonegmn1l  38262  prnc  38388  reuopreuprim  47986  uhgrimisgrgriclem  48406