Theorem anim12 806

Description: Conjoin antecedents and consequents of two premises. This is the closed theorem form of anim12d 609. 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
anim12	⊢ (((𝜑 → 𝜓) ∧ (𝜒 → 𝜃)) → ((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜃)))

Proof of Theorem anim12

Step	Hyp	Ref	Expression
1		id 22	. 2 ⊢ ((𝜑 → 𝜓) → (𝜑 → 𝜓))
2		id 22	. 2 ⊢ ((𝜒 → 𝜃) → (𝜒 → 𝜃))
3	1, 2	im2anan9 620	1 ⊢ (((𝜑 → 𝜓) ∧ (𝜒 → 𝜃)) → ((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜃)))

Syntax hints:   → wi 4   ∧ wa 396

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 206  df-an 397

This theorem is referenced by:  euind  3659  reuind  3688  reusv3i  5327  opelopabt  5445  wemaplem2  9306  rexanre  15058  rlimcn3  15299  o1of2  15322  o1rlimmul  15328  2sqlem6  26571  spanuni  29906  bj-nnfan  34930  isbasisrelowllem1  35526  isbasisrelowllem2  35527  heicant  35812  pm11.71  42015