Theorem sylanb 458

Description: A syllogism inference. (Contributed by NM, 18-May-1994.)

Hypotheses

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

Assertion

Ref	Expression
sylanb	|- ((ph /\ ch) -> th)

Proof of Theorem sylanb

Step	Hyp	Ref	Expression
1		sylanb.1	. . 3 |- (ph <-> ps)
2	1	biimpi 186	. 2 |- (ph -> ps)
3		sylanb.2	. 2 |- ((ps /\ ch) -> th)
4	2, 3	sylan 457	1 |- ((ph /\ ch) -> th)

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358

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 177  df-an 360

This theorem is referenced by:  syl2anb  465  anabsan  786  eqtr2  2371  pm13.181  2590  rmob  3135  sspsstr  3375  disjne  3597  xpcan2  5059  fssres  5239  funbrfvb  5361  fvco  5384  fvimacnvi  5403  ffvresb  5432  leaddc2  6216  lemuc2  6255