Theorem mpani 657

Description: An inference based on modus ponens. (Contributed by NM, 10-Apr-1994.) (Proof shortened by Wolf Lammen, 19-Nov-2012.)

Hypotheses

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

Assertion

Ref	Expression
mpani	⊢ (φ → (χ → θ))

Proof of Theorem mpani

Step	Hyp	Ref	Expression
1		mpani.1	. . 3 ⊢ ψ
2	1	a1i 10	. 2 ⊢ (φ → ψ)
3		mpani.2	. 2 ⊢ (φ → ((ψ ∧ χ) → θ))
4	2, 3	mpand 656	1 ⊢ (φ → (χ → θ))

Syntax hints:   → wi 4   ∧ wa 358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

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

This theorem is referenced by:  mp2ani  659