Theorem mpii 46

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 45	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:  intmin  4944  dfiin2g  5008  ssorduni  7773  sucexeloniOLD  7804  suceloniOLD  7806  lublecllem  18370  irredmul  20389  opnneiid  23064  isufil2  23846  mdbr3  32278  mdbr4  32279  dmdbr5  32289  filnetlem4  36399  iunord  49540