Theorem mpid 42

Description: A nested modus ponens deduction. (Contributed by NM, 14-Dec-2004.)

Hypotheses

Ref	Expression
mpid.1	|- ( ph -> ch )
mpid.2	|- ( ph -> ( ps -> ( ch -> th ) ) )

Assertion

Ref	Expression
mpid	|- ( ph -> ( ps -> th ) )

Proof of Theorem mpid

Step	Hyp	Ref	Expression
1		mpid.1	. . 3 |- ( ph -> ch )
2	1	a1d 22	. 2 |- ( ph -> ( ps -> ch ) )
3		mpid.2	. 2 |- ( ph -> ( ps -> ( ch -> th ) ) )
4	2, 3	mpdd 41	1 |- ( ph -> ( ps -> th ) )

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:  mp2d  47  pm2.43a  51  embantd  56  mpan2d  432  ceqsalt  2848  rspcimdv  2930  fvimacnv  5824  riotass2  6067  pr2ne  7538  0mnnnnn0  9599  caucvgre  11761  climcn1  12090  climcn2  12091  gcdaddm  12777  dvdsgcd  12805  coprmgcdb  12882  nprm  12917  pcqmul  13102  grpid  13893  uniopn  15151  metcnp3  15661  cncfco  15741  eupth2fi  16818