Theorem mp2ani 659

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

Hypotheses

Ref	Expression
mp2ani.1	⊢ ψ
mp2ani.2	⊢ χ
mp2ani.3	⊢ (φ → ((ψ ∧ χ) → θ))

Assertion

Ref	Expression
mp2ani	⊢ (φ → θ)

Proof of Theorem mp2ani

Step	Hyp	Ref	Expression
1		mp2ani.2	. 2 ⊢ χ
2		mp2ani.1	. . 3 ⊢ ψ
3		mp2ani.3	. . 3 ⊢ (φ → ((ψ ∧ χ) → θ))
4	2, 3	mpani 657	. 2 ⊢ (φ → (χ → θ))
5	1, 4	mpi 16	1 ⊢ (φ → 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: (None)