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  425  ceqsalt  2751  rspcimdv  2830  fvimacnv  5599  riotass2  5823  pr2ne  7144  0mnnnnn0  9142  caucvgre  10919  climcn1  11245  climcn2  11246  gcdaddm  11913  dvdsgcd  11941  coprmgcdb  12016  nprm  12051  pcqmul  12231  uniopn  12599  metcnp3  13111  cncfco  13178