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:  fisseneq  7038  exmidapne  7379  prloc  7611  axcaucvglemres  8019  seqf1oglem1  10671  seqf1oglem2  10672  bezoutlemmain  12363  coprm  12510  sqrt2irr  12528  oddprmdvds  12721  lmodfopnelem1  14130  xblss2ps  14920  xblss2  14921  perfectlem2  15516  lgsprme0  15563  pw1nct  16014  apdiff  16061