Theorem prth 554

Description: Conjoin antecedents and consequents of two premises. This is the closed theorem form of anim12d 546. Theorem *3.47 of [WhiteheadRussell] p. 113. It was proved by Leibniz, and it evidently pleased him enough to call it praeclarum theorema (splendid theorem). (Contributed by NM, 12-Aug-1993.) (Proof shortened by Wolf Lammen, 7-Apr-2013.)

Assertion

Ref	Expression
prth	⊢ (((φ → ψ) ∧ (χ → θ)) → ((φ ∧ χ) → (ψ ∧ θ)))

Proof of Theorem prth

Step	Hyp	Ref	Expression
1		simpl 443	. 2 ⊢ (((φ → ψ) ∧ (χ → θ)) → (φ → ψ))
2		simpr 447	. 2 ⊢ (((φ → ψ) ∧ (χ → θ)) → (χ → θ))
3	1, 2	anim12d 546	1 ⊢ (((φ → ψ) ∧ (χ → θ)) → ((φ ∧ χ) → (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:  mo  2226  2mo  2282  euind  3024  reuind  3040  reuss2  3536  opelopabt  4700