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  2789  rspcimdv  2869  fvimacnv  5677  riotass2  5904  pr2ne  7259  0mnnnnn0  9281  caucvgre  11146  climcn1  11473  climcn2  11474  gcdaddm  12151  dvdsgcd  12179  coprmgcdb  12256  nprm  12291  pcqmul  12472  grpid  13171  uniopn  14237  metcnp3  14747  cncfco  14827