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

Theorem adantrrl 721
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
adantrrl ((𝜑 ∧ (𝜓 ∧ (𝜏𝜒))) → 𝜃)

Proof of Theorem adantrrl
StepHypRef Expression
1 simpr 485 . 2 ((𝜏𝜒) → 𝜒)
2 adantr2.1 . 2 ((𝜑 ∧ (𝜓𝜒)) → 𝜃)
31, 2sylanr2 680 1 ((𝜑 ∧ (𝜓 ∧ (𝜏𝜒))) → 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 206  df-an 397
This theorem is referenced by:  zorn2lem6  10358  ltmul12a  11932  mndind  18563  neiint  22361  neissex  22384  1stcfb  22702  1stcrest  22710  grporcan  29168  mdslmd3i  30982  colineardim1  34459  cvratlem  37689  ps-2  37746  fsuppssind  40542
  Copyright terms: Public domain W3C validator