Theorem mpii 46

Description: A doubly nested modus ponens inference. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Wolf Lammen, 31-Jul-2012.)

Hypotheses

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

Assertion

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

Proof of Theorem mpii

Step	Hyp	Ref	Expression
1		mpii.1	. . 3 |- ch
2	1	a1i 11	. 2 |- ( ps -> ch )
3		mpii.2	. 2 |- ( ph -> ( ps -> ( ch -> th ) ) )
4	2, 3	mpdi 45	1 |- ( ph -> ( ps -> th ) )

Syntax hints:   -> wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7

This theorem is referenced by:  intmin  4497  dfiin2g  4553  ssorduni  6985  suceloni  7013  lublecllem  16988  irredmul  18709  opnneiid  20930  isufil2  21712  mdbr3  29156  mdbr4  29157  dmdbr5  29167  filnetlem4  32376  iunord  42422