Theorem imp31 256

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

Hypothesis

Ref	Expression
imp3.1	⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))

Assertion

Ref	Expression
imp31	⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃)

Proof of Theorem imp31

Step	Hyp	Ref	Expression
1		imp3.1	. . 3 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))
2	1	imp 124	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝜒 → 𝜃))
3	2	imp 124	1 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃)

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:  imp41  353  imp5d  359  impl  380  anassrs  400  an31s  570  con4biddc  858  3imp  1195  3expa  1205  bilukdc  1407  reusv3  4495  dfimafn  5609  funimass4  5611  funimass3  5678  isopolem  5869  smores2  6352  tfrlem9  6377  nnmordi  6574  mulcanpig  7402  elnnz  9336  nzadd  9378  irradd  9720  irrmul  9721  uzsubsubfz  10122  fzo1fzo0n0  10259  elfzonelfzo  10306  infpnlem1  12528  tgcl  14300