Theorem mpii 47

Description: A doubly nested modus ponens inference. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Wolf Lammen, 31-Jul-2012.)

Hypotheses

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

Assertion

Ref	Expression
mpii	⊢ (𝜑 → (𝜓 → 𝜃))

Proof of Theorem mpii

Step	Hyp	Ref	Expression
1		mpii.1	. . 3 ⊢ 𝜒
2	1	a1i 11	. 2 ⊢ (𝜓 → 𝜒)
3		mpii.2	. 2 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))
4	2, 3	mpdi 46	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:  intmin  4927  ssorduni  7793  lublecllem  18537  irredmul  20664  opnneiid  23449  isufil2  24232  mdbr3  32904  mdbr4  32905  dmdbr5  32915  mirimanoffr  36546  filnetlem4  37261  iunord  50952