Theorem imp32 257

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

Hypothesis

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

Assertion

Ref	Expression
imp32	⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃)

Proof of Theorem imp32

Step	Hyp	Ref	Expression
1		imp3.1	. . 3 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))
2	1	impd 254	. 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:  imp42  354  impr  379  anasss  399  an13s  569  3expb  1230  reuss2  3487  reupick  3491  po2nr  4406  fvmptt  5738  fliftfund  5937  f1ocnv2d  6226  addclpi  7546  addnidpig  7555  mulnqprl  7787  mulnqpru  7788  ltsubrp  9924  ltaddrp  9925  pfxccat3  11314  divgcdcoprm0  12672  infpnlem1  12931  imasmnd2  13534  imasgrp2  13696  imasrng  13968  imasring  14076  innei  14886  tgcnp  14932  isxmetd  15070  2lgslem1a1  15814