Theorem syldan 456

Description: A syllogism deduction with conjoined antecedents. (Contributed by NM, 24-Feb-2005.) (Proof shortened by Wolf Lammen, 6-Apr-2013.)

Hypotheses

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

Assertion

Ref	Expression
syldan	⊢ ((φ ∧ ψ) → θ)

Proof of Theorem syldan

Step	Hyp	Ref	Expression
1		syldan.1	. 2 ⊢ ((φ ∧ ψ) → χ)
2		syldan.2	. . . 4 ⊢ ((φ ∧ χ) → θ)
3	2	expcom 424	. . 3 ⊢ (χ → (φ → θ))
4	3	adantrd 454	. 2 ⊢ (χ → ((φ ∧ ψ) → θ))
5	1, 4	mpcom 32	1 ⊢ ((φ ∧ ψ) → 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:  sylan2  460  sbcied2  3084  csbied2  3180  lefinaddc  4451  vfinncvntsp  4550  addlec  6209  nchoicelem5  6294