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  7334  isotilem  7347  recexprlemss1l  8003  recexprlemss1u  8004  caucvgsrlemoffres  8168  suplocsrlem  8176  nnindnn  8261  eqord1  8813  lemul12b  9194  lt2msq  9219  lbreu  9278  cju  9294  nnind  9323  uz11  9955  xrre2  10234  ico0  10707  ioc0  10708  expcan  11170  swrdccatin2  11517  gcdaddm  12780  bezoutlemaz  12799  bezoutlembz  12800  isprm3  12915  prmdiveq  13037  mulgpropdg  14020  imasabl  14224  subrgdvds  14627  epttop  15282  cnptopresti  15430  cnptoprest  15431  txcnp  15463  addcncntoplem  15753  mulcncflem  15799  umgrvad2edg  16618  wlk1walkdom  16766  bj-stand  16942  exmidsbthrlem  17233