Theorem prth 587

Description: Conjoin antecedents and consequents of two premises. This is the closed theorem form of anim12d 578. 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 466	. 2 ⊢ (((𝜑 → 𝜓) ∧ (𝜒 → 𝜃)) → (𝜑 → 𝜓))
2		simpr 470	. 2 ⊢ (((𝜑 → 𝜓) ∧ (𝜒 → 𝜃)) → (𝜒 → 𝜃))
3	1, 2	anim12d 578	1 ⊢ (((𝜑 → 𝜓) ∧ (𝜒 → 𝜃)) → ((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜃)))

Syntax hints:   → wi 4   ∧ wa 378

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 192  df-an 380

This theorem is referenced by:  euind  3249  reuind  3267  reuss2  3749  reusv3i  4649  opelopabt  4754  wemaplem2  8145  rexanre  13569  rlimcn2  13814  o1of2  13836  o1rlimmul  13842  2sqlem6  24458  spanuni  27360  bj-mo3OLD  31631  isbasisrelowllem1  31979  isbasisrelowllem2  31980  heicant  32213  pm11.71  37104