Theorem syl3an2 1312

Description: A syllogism inference. (Contributed by NM, 22-Aug-1995.)

Hypotheses

Ref	Expression
syl3an2.1	|- ( ph -> ch )
syl3an2.2	|- ( ( ps /\ ch /\ th ) -> ta )

Assertion

Ref	Expression
syl3an2	|- ( ( ps /\ ph /\ th ) -> ta )

Proof of Theorem syl3an2

Step	Hyp	Ref	Expression
1		syl3an2.1	. . 3 |- ( ph -> ch )
2		syl3an2.2	. . . 4 |- ( ( ps /\ ch /\ th ) -> ta )
3	2	3exp 1233	. . 3 |- ( ps -> ( ch -> ( th -> ta ) ) )
4	1, 3	syl5 32	. 2 |- ( ps -> ( ph -> ( th -> ta ) ) )
5	4	3imp 1224	1 |- ( ( ps /\ ph /\ th ) -> ta )

This proof depends on syntax axioms:   -> wi 4   /\ w3a 1009

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  df-3an 1011

This theorem is used by:  syl3an2b  1315  syl3an2br  1318  syl3anl2  1327  nndi  6759  nnmass  6760  prarloclemarch2  7786  1idprl  7957  1idpru  7958  recexprlem1ssl  8000  recexprlem1ssu  8001  msqge0  8944  mulge0  8947  divsubdirap  9038  divdiv32ap  9050  peano2uz  9983  fzoshftral  10657  expdivap  11027  bcval5  11201  ccats1val1g  11407  redivap  11639  imdivap  11646  absdiflt  11858  absdifle  11859  retanclap  12489  tannegap  12495  lcmgcdeq  12861  isprm3  12896  prmdvdsexpb  12927  dvdsprmpweqnn  13115  mulgaddcomlem  13948  mulginvcom  13950  cnpf2  15308  blres  15535