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  4899  dfiin2g  4962  ssorduni  7629  sucexeloni  7658  suceloniOLD  7660  lublecllem  18078  irredmul  19951  opnneiid  22277  isufil2  23059  mdbr3  30659  mdbr4  30660  dmdbr5  30670  filnetlem4  34570  iunord  46382