MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  adantrlr Structured version   Visualization version   GIF version

Theorem adantrlr 719
Description: Deduction adding a conjunct to antecedent. (Contributed by NM, 26-Dec-2004.) (Proof shortened by Wolf Lammen, 4-Dec-2012.)
Hypothesis
Ref Expression
adantr2.1 ((𝜑 ∧ (𝜓𝜒)) → 𝜃)
Assertion
Ref Expression
adantrlr ((𝜑 ∧ ((𝜓𝜏) ∧ 𝜒)) → 𝜃)

Proof of Theorem adantrlr
StepHypRef Expression
1 simpl 483 . 2 ((𝜓𝜏) → 𝜓)
2 adantr2.1 . 2 ((𝜑 ∧ (𝜓𝜒)) → 𝜃)
31, 2sylanr1 678 1 ((𝜑 ∧ ((𝜓𝜏) ∧ 𝜒)) → 𝜃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396
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 208  df-an 397
This theorem is referenced by:  smoord  7996  addsrmo  10487  mulsrmo  10488  lediv12a  11525  nrmmetd  23099  pntrmax  26054  ablo4  28241  mdslmd3i  30023  atom1d  30044  fsumiunle  30459  esumiun  31239  poimirlem28  34787  fdc  34888  incsequz  34891  crngm4  35149  ps-2  36481  aacllem  44730
  Copyright terms: Public domain W3C validator