Theorem mpid 42

Description: A nested modus ponens deduction. (Contributed by NM, 14-Dec-2004.)

Hypotheses

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

Assertion

Ref	Expression
mpid	⊢ (𝜑 → (𝜓 → 𝜃))

Proof of Theorem mpid

Step	Hyp	Ref	Expression
1		mpid.1	. . 3 ⊢ (𝜑 → 𝜒)
2	1	a1d 22	. 2 ⊢ (𝜑 → (𝜓 → 𝜒))
3		mpid.2	. 2 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))
4	2, 3	mpdd 41	1 ⊢ (𝜑 → (𝜓 → 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:  mp2d  47  pm2.43a  51  embantd  56  mpan2d  428  ceqsalt  2842  rspcimdv  2924  fvimacnv  5798  riotass2  6040  pr2ne  7502  0mnnnnn0  9548  caucvgre  11694  climcn1  12021  climcn2  12022  gcdaddm  12708  dvdsgcd  12736  coprmgcdb  12813  nprm  12848  pcqmul  13029  grpid  13797  uniopn  14995  metcnp3  15505  cncfco  15585  eupth2fi  16603