Theorem mpdi 43

Description: A nested modus ponens deduction. (Contributed by NM, 16-Apr-2005.) (Proof shortened by O'Cat, 15-Jan-2008.)

Hypotheses

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

Assertion

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

Proof of Theorem mpdi

Step	Hyp	Ref	Expression
1		mpdi.1	. . 3 |- ( ps -> ch )
2	1	a1i 9	. 2 |- ( ph -> ( ps -> ch ) )
3		mpdi.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:  mpii  44  pm2.43d  50  gencbval  2871  sbcimdv  3117  suctr  4566  suppssdc  6500  tfrlem9  6590  lbzbi  10016  flqeqceilz  10755  ndvdsadd  12698  gcdneg  12759  bj-inf2vnlem2  16997