Theorem mp2 16

Description: A double modus ponens inference. (Contributed by NM, 5-Apr-1994.) (Proof shortened by Wolf Lammen, 23-Jul-2013.)

Hypotheses

Ref	Expression
mp2.1	|- ph
mp2.2	|- ps
mp2.3	|- ( ph -> ( ps -> ch ) )

Assertion

Ref	Expression
mp2	|- ch

Proof of Theorem mp2

Step	Hyp	Ref	Expression
1		mp2.1	. 2 |- ph
2		mp2.2	. . 3 |- ps
3		mp2.3	. . 3 |- ( ph -> ( ps -> ch ) )
4	2, 3	mpi 15	. 2 |- ( ph -> ch )
5	1, 4	ax-mp 5	1 |- ch

This proof depends on syntax axioms:   -> wi 4

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7

This theorem is used by:  impbii  126  pm3.2i  272  sstri  3257  0disj  4128  disjx0  4130  ontr2exmid  4674  0elsucexmid  4714  relres  5093  cnvdif  5197  funopab4  5417  fun0  5442  fvsn  5913  reltpos  6526  tpostpos  6540  tpos0  6550  oawordriexmid  6748  swoer  6840  xpider  6885  erinxp  6888  domfiexmid  7188  diffitest  7197  pw1dom2  7592  ltrel  8393  lerel  8395  frecfzennn  10883  sum0  12179  qnnen  13379  hovercncf  15889  lgsquadlem1  16418  lgsquadlem2  16419  usgrexmpldifpr  16712