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))

This proof depends on syntax axioms:   → wi 4

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7

This theorem is used by:  mp2d  47  pm2.43a  51  embantd  56  mpan2d  432  ceqsalt  2848  rspcimdv  2930  fvimacnv  5827  riotass2  6070  pr2ne  7545  0mnnnnn0  9607  caucvgre  11770  climcn1  12100  climcn2  12101  gcdaddm  12787  dvdsgcd  12815  coprmgcdb  12892  nprm  12927  pcqmul  13112  grpid  13904  uniopn  15239  metcnp3  15749  cncfco  15829  eupth2fi  16937