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  5991  fisseneq  7121  exmidapne  7472  prloc  7704  axcaucvglemres  8112  seqf1oglem1  10774  seqf1oglem2  10775  wrdind  11296  wrd2ind  11297  bezoutlemmain  12562  coprm  12709  sqrt2irr  12727  oddprmdvds  12920  lmodfopnelem1  14331  xblss2ps  15121  xblss2  15122  perfectlem2  15717  lgsprme0  15764  pw1nct  16554  apdiff  16602