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  2786  rspcimdv  2866  fvimacnv  5674  riotass2  5901  pr2ne  7254  0mnnnnn0  9275  caucvgre  11128  climcn1  11454  climcn2  11455  gcdaddm  12124  dvdsgcd  12152  coprmgcdb  12229  nprm  12264  pcqmul  12444  grpid  13114  uniopn  14180  metcnp3  14690  cncfco  14770