Theorem anim12d 335

Description: Conjoin antecedents and consequents in a deduction. (Contributed by NM, 3-Apr-1994.) (Proof shortened by Wolf Lammen, 18-Dec-2013.)

Hypotheses

Ref	Expression
anim12d.1	|- ( ph -> ( ps -> ch ) )
anim12d.2	|- ( ph -> ( th -> ta ) )

Assertion

Ref	Expression
anim12d	|- ( ph -> ( ( ps /\ th ) -> ( ch /\ ta ) ) )

Proof of Theorem anim12d

Step	Hyp	Ref	Expression
1		anim12d.1	. 2 |- ( ph -> ( ps -> ch ) )
2		anim12d.2	. 2 |- ( ph -> ( th -> ta ) )
3		idd 21	. 2 |- ( ph -> ( ( ch /\ ta ) -> ( ch /\ ta ) ) )
4	1, 2, 3	syl2and 295	1 |- ( ph -> ( ( ps /\ th ) -> ( ch /\ ta ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This proof depends on definitions:  df-bi 117

This theorem is used by:  anim1d  336  anim2d  337  anim12  344  im2anan9  606  anim12dan  608  3anim123d  1360  hband  1542  hbbid  1628  spsbim  1896  moim  2151  moimv  2153  2euswapdc  2178  rspcimedv  2931  soss  4459  trin2  5179  xp11m  5226  funss  5396  fun  5561  dff13  5974  f1eqcocnv  5997  isores3  6021  isosolem  6030  f1o2ndf1  6464  tposfn2  6537  tposf1o2  6541  nndifsnid  6780  nnaordex  6801  supmoti  7333  isotilem  7346  recexprlemss1l  8002  recexprlemss1u  8003  caucvgsrlemoffres  8167  suplocsrlem  8175  nnindnn  8260  eqord1  8811  lemul12b  9191  lt2msq  9216  lbreu  9275  cju  9291  nnind  9320  uz11  9945  xrre2  10223  ico0  10696  ioc0  10697  expcan  11154  swrdccatin2  11501  gcdaddm  12761  bezoutlemaz  12780  bezoutlembz  12781  isprm3  12896  prmdiveq  13014  mulgpropdg  13967  imasabl  14140  subrgdvds  14543  epttop  15191  cnptopresti  15339  cnptoprest  15340  txcnp  15372  addcncntoplem  15662  mulcncflem  15708  umgrvad2edg  16452  wlk1walkdom  16600  bj-stand  16776  exmidsbthrlem  17067