Theorem mp2and 660

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

Hypotheses

Ref	Expression
mp2and.1	⊢ (φ → ψ)
mp2and.2	⊢ (φ → χ)
mp2and.3	⊢ (φ → ((ψ ∧ χ) → θ))

Assertion

Ref	Expression
mp2and	⊢ (φ → θ)

Proof of Theorem mp2and

Step	Hyp	Ref	Expression
1		mp2and.2	. 2 ⊢ (φ → χ)
2		mp2and.1	. . 3 ⊢ (φ → ψ)
3		mp2and.3	. . 3 ⊢ (φ → ((ψ ∧ χ) → θ))
4	2, 3	mpand 656	. 2 ⊢ (φ → (χ → θ))
5	1, 4	mpd 14	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:  ssnelpssd  3615  nnsucelr  4429  sfinltfin  4536  vfinncvntnn  4549  trd  5922  frd  5923  antid  5930