Theorem imp42 425

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 417	. 2 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → (𝜃 → 𝜏))
3	2	imp 405	1 ⊢ (((𝜑 ∧ (𝜓 ∧ 𝜒)) ∧ 𝜃) → 𝜏)

Syntax hints:   → wi 4   ∧ wa 394

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 206  df-an 395

This theorem is referenced by:  imp55  441  ltexprlem7  11076  iscatd  17681  isposd  18343  pospropd  18347  mulgghm2  21462  ordtbaslem  23180  txbas  23559  nocvxminlem  27804  frgrncvvdeqlem8  30236  grporcan  30448  chirredlem1  32320  cvxpconn  35083  cvxsconn  35084  rngonegmn1l  37655  prnc  37781  reuopreuprim  47134  uhgrimisgrgriclem  47513