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  6030  fisseneq  7197  exmidapne  7579  prloc  7811  axcaucvglemres  8219  seqf1oglem1  10888  seqf1oglem2  10889  wrdind  11422  wrd2ind  11423  bezoutlemmain  12702  coprm  12849  sqrt2irr  12867  oddprmdvds  13060  lmodfopnelem1  14521  xblss2ps  15318  xblss2  15319  perfectlem2  15917  lgsprme0  15964  pw1nct  16826  apdiff  16881