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Theorem im2anan9 808
Description: Deduction joining nested implications to form implication of conjunctions. (Contributed by NM, 29-Feb-1996.)
Hypotheses
Ref Expression
im2an9.1  |-  ( ph  ->  ( ps  ->  ch ) )
im2an9.2  |-  ( th 
->  ( ta  ->  et ) )
Assertion
Ref Expression
im2anan9  |-  ( (
ph  /\  th )  ->  ( ( ps  /\  ta )  ->  ( ch 
/\  et ) ) )

Proof of Theorem im2anan9
StepHypRef Expression
1 im2an9.1 . . 3  |-  ( ph  ->  ( ps  ->  ch ) )
21adantr 451 . 2  |-  ( (
ph  /\  th )  ->  ( ps  ->  ch ) )
3 im2an9.2 . . 3  |-  ( th 
->  ( ta  ->  et ) )
43adantl 452 . 2  |-  ( (
ph  /\  th )  ->  ( ta  ->  et ) )
52, 4anim12d 546 1  |-  ( (
ph  /\  th )  ->  ( ( ps  /\  ta )  ->  ( ch 
/\  et ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358
This theorem is referenced by:  im2anan9r  809  ax11eq  2145  ax11el  2146  trin  4139  somo  4364  xpss12  4808  f1oun  5508  poxp  6243  soxp  6244  brecop  6767  ingru  8453  genpss  8644  genpnnp  8645  tgcl  16723  txlm  17358  oriso  25317  ridlideq  25438  limptlimpr2lem2  25678  lvsovso  25729  claddrvr  25751  distmlva  25791  icccon2  25803
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 177  df-an 360
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