Theorem mpan2d 655

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

Hypotheses

Ref	Expression
mpan2d.1	⊢ (φ → χ)
mpan2d.2	⊢ (φ → ((ψ ∧ χ) → θ))

Assertion

Ref	Expression
mpan2d	⊢ (φ → (ψ → θ))

Proof of Theorem mpan2d

Step	Hyp	Ref	Expression
1		mpan2d.1	. 2 ⊢ (φ → χ)
2		mpan2d.2	. . 3 ⊢ (φ → ((ψ ∧ χ) → θ))
3	2	exp3a 425	. 2 ⊢ (φ → (ψ → (χ → θ)))
4	1, 3	mpid 37	1 ⊢ (φ → (ψ → θ))

This proof depends on syntax axioms:   → wi 4   ∧ wa 358

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-an 360

This theorem is used by:  mpand  656  mpan2i  658  ltfintri  4467  leconnnc  6219  nclenn  6250  ncslesuc  6268  nchoicelem4  6293