Theorem mp2d 47

Description: A double modus ponens deduction. (Contributed by NM, 23-May-2013.) (Proof shortened by Wolf Lammen, 23-Jul-2013.)

Hypotheses

Ref	Expression
mp2d.1	⊢ (𝜑 → 𝜓)
mp2d.2	⊢ (𝜑 → 𝜒)
mp2d.3	⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))

Assertion

Ref	Expression
mp2d	⊢ (𝜑 → 𝜃)

Proof of Theorem mp2d

Step	Hyp	Ref	Expression
1		mp2d.1	. 2 ⊢ (𝜑 → 𝜓)
2		mp2d.2	. . 3 ⊢ (𝜑 → 𝜒)
3		mp2d.3	. . 3 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))
4	2, 3	mpid 42	. 2 ⊢ (𝜑 → (𝜓 → 𝜃))
5	1, 4	mpd 13	1 ⊢ (𝜑 → 𝜃)

Syntax hints:   → wi 4

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

This theorem is referenced by:  riotaeqimp  6057  fisseneq  7236  exmidapne  7620  prloc  7852  axcaucvglemres  8260  seqf1oglem1  10939  seqf1oglem2  10940  wrdind  11477  wrd2ind  11478  bezoutlemmain  12758  coprm  12905  sqrt2irr  12923  oddprmdvds  13116  lmodfopnelem1  14644  xblss2ps  15488  xblss2  15489  perfectlem2  16097  lgsprme0  16144  dichmul0orlem7  16742  pw1nct  17016  apdiff  17071