Theorem 3adantr3 1173

Description: Deduction adding a conjunct to antecedent. (Contributed by NM, 27-Apr-2005.)

Hypothesis

Ref	Expression
3adantr.1	⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃)

Assertion

Ref	Expression
3adantr3	⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜏)) → 𝜃)

Proof of Theorem 3adantr3

Step	Hyp	Ref	Expression
1		3simpa 1149	. 2 ⊢ ((𝜓 ∧ 𝜒 ∧ 𝜏) → (𝜓 ∧ 𝜒))
2		3adantr.1	. 2 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → 𝜃)
3	1, 2	sylan2 594	1 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜏)) → 𝜃)

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:  3adant3r3  1186  3ad2antr1  1190  3ad2antr2  1191  sotr2  5574  dfwe2  7729  smogt  8309  infsupprpr  9421  wlogle  11682  fzadd2  13487  swrdspsleq  14601  tanadd  16104  prdssgrpd  18670  prdsmndd  18707  mhmmnd  19006  imasrng  20124  imasring  20278  prdslmodd  20932  sraassab  21835  mpllsslem  21967  scmatlss  22481  mdetunilem3  22570  ptclsg  23571  tmdgsum2  24052  isxmet2d  24283  xmetres2  24317  prdsxmetlem  24324  comet  24469  iimulcl  24901  icoopnst  24904  iocopnst  24905  icccvx  24916  dvfsumrlim  26006  dvfsumrlim2  26007  colhp  28854  eengtrkg  29071  wwlksnredwwlkn  29980  dmdsl3  32402  eqgvscpbl  33442  resconn  35459  poimirlem28  37896  poimirlem32  37900  broucube  37902  ftc1anclem7  37947  ftc1anc  37949  isdrngo2  38206  iscringd  38246  unichnidl  38279  lplnle  39913  2llnjN  39940  2lplnj  39993  osumcllem11N  40339  cdleme1  40600  erngplus2  41177  erngplus2-rN  41185  erngdvlem3  41363  erngdvlem3-rN  41371  dvaplusgv  41383  dvalveclem  41398  dvhvaddass  41470  dvhlveclem  41481  dihmeetlem12N  41691  issmflem  47082  fmtnoprmfac1  47922  lincresunit3lem2  48837  lincresunit3  48838