Theorem mpdd 41

Description: A nested modus ponens deduction. (Contributed by NM, 12-Dec-2004.)

Hypotheses

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

Assertion

Ref	Expression
mpdd	⊢ (𝜑 → (𝜓 → 𝜃))

Proof of Theorem mpdd

Step	Hyp	Ref	Expression
1		mpdd.1	. 2 ⊢ (𝜑 → (𝜓 → 𝜒))
2		mpdd.2	. . 3 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))
3	2	a2d 26	. 2 ⊢ (𝜑 → ((𝜓 → 𝜒) → (𝜓 → 𝜃)))
4	1, 3	mpd 13	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:  mpid  42  mpdi  43  syld  45  syl6c  66  mpteqb  5664  oprabid  5966  nnmordi  6592  nnmord  6593  brecop  6702  findcard2  6968  findcard2s  6969  ordiso2  7119  zindd  9473  cau3lem  11344  climcau  11577  dvdsabseq  12077  znrrg  14340  metrest  14896  bj-charfunr  15610