Theorem 3anbi13d 1393

Description: Deduction conjoining and adding a conjunct to equivalences. (Contributed by NM, 8-Sep-2006.)

Hypotheses

Ref	Expression
3anbi12d.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))
3anbi12d.2	⊢ (𝜑 → (𝜃 ↔ 𝜏))

Assertion

Ref	Expression
3anbi13d	⊢ (𝜑 → ((𝜓 ∧ 𝜂 ∧ 𝜃) ↔ (𝜒 ∧ 𝜂 ∧ 𝜏)))

Proof of Theorem 3anbi13d

Step	Hyp	Ref	Expression
1		3anbi12d.1	. 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
2		biidd 251	. 2 ⊢ (𝜑 → (𝜂 ↔ 𝜂))
3		3anbi12d.2	. 2 ⊢ (𝜑 → (𝜃 ↔ 𝜏))
4	1, 2, 3	3anbi123d 1391	1 ⊢ (𝜑 → ((𝜓 ∧ 𝜂 ∧ 𝜃) ↔ (𝜒 ∧ 𝜂 ∧ 𝜏)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ w3a 1031

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 196  df-an 385  df-3an 1033

This theorem is referenced by:  3anbi3d  1397  ax12wdemo  1999  f1dom3el3dif  6426  wfrlem1  7301  wfrlem3a  7304  wfrlem15  7316  cofsmo  8974  axdc3lem3  9157  axdc3lem4  9158  iscatd2  16165  psgnunilem1  17736  nn0gsumfz  18203  opprsubrg  18624  lsspropd  18838  mdetunilem3  20239  mdetunilem9  20245  smadiadetr  20300  lmres  20914  cnhaus  20968  regsep2  20990  dishaus  20996  ordthauslem  20997  nconsubb  21036  pthaus  21251  txhaus  21260  xkohaus  21266  regr1lem  21352  ustval  21816  methaus  22135  metnrmlem3  22472  pmltpclem1  23024  axtgeucl  25171  iscgrad  25503  dfcgra2  25521  f1otrge  25552  axeuclidlem  25642  2wlkonot  26392  2spthonot  26393  usg2spthonot0  26416  vdn0frgrav2  26550  vdgn0frgrav2  26551  vdn1frgrav2  26552  vdgn1frgrav2  26553  ex-opab  26681  isnvlem  26849  ajval  27101  adjeu  28132  adjval  28133  adj1  28176  adjeq  28178  cnlnssadj  28323  br8d  28802  lt2addrd  28903  xlt2addrd  28913  measval  29588  br8  30899  br6  30900  br4  30901  nofulllem5  31105  brcgr3  31323  brsegle  31385  fvray  31418  linedegen  31420  fvline  31421  poimirlem28  32607  isopos  33485  hlsuprexch  33685  2llnjN  33871  2lplnj  33924  cdlemk42  35247  zindbi  36529  jm2.27  36593  stoweidlem43  38936  fourierdlem42  39042  umgrvad2edg  40440  elwspths2spth  41171  upgr3v3e3cycl  41347  upgr4cycl4dv4e  41352  vdgn1frgrv2  41466