MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  3adantr1 Structured version   Visualization version   GIF version

Theorem 3adantr1 1170
Description: Deduction adding a conjunct to antecedent. (Contributed by NM, 27-Apr-2005.)
Hypothesis
Ref Expression
3adantr.1 ((𝜑 ∧ (𝜓𝜒)) → 𝜃)
Assertion
Ref Expression
3adantr1 ((𝜑 ∧ (𝜏𝜓𝜒)) → 𝜃)

Proof of Theorem 3adantr1
StepHypRef Expression
1 3simpc 1151 . 2 ((𝜏𝜓𝜒) → (𝜓𝜒))
2 3adantr.1 . 2 ((𝜑 ∧ (𝜓𝜒)) → 𝜃)
31, 2sylan2 593 1 ((𝜑 ∧ (𝜏𝜓𝜒)) → 𝜃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087
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 207  df-an 396  df-3an 1089
This theorem is referenced by:  3adant3r1  1183  3ad2antr3  1191  swopo  5603  omeulem1  8620  divmuldiv  11967  imasmnd2  18787  imasgrp2  19073  imasrng  20174  srgbinomlem2  20224  imasring  20327  abvdiv  20830  mdetunilem9  22626  lly1stc  23504  icccvx  24981  dchrpt  27311  dipsubdir  30867  poimirlem4  37631  fdc  37752  unichnidl  38038  dmncan1  38083  pexmidlem6N  39977  erngdvlem3  40992  erngdvlem3-rN  41000  dvalveclem  41027  dvhvaddass  41099  dvhlveclem  41110  issmflem  46742  prproropf1olem3  47492
  Copyright terms: Public domain W3C validator