Theorem prth 595

Description: Conjoin antecedents and consequents of two premises. This is the closed theorem form of anim12d 586. 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 473	. 2 ⊢ (((𝜑 → 𝜓) ∧ (𝜒 → 𝜃)) → (𝜑 → 𝜓))
2		simpr 477	. 2 ⊢ (((𝜑 → 𝜓) ∧ (𝜒 → 𝜃)) → (𝜒 → 𝜃))
3	1, 2	anim12d 586	1 ⊢ (((𝜑 → 𝜓) ∧ (𝜒 → 𝜃)) → ((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜃)))

Syntax hints:   → wi 4   ∧ wa 384

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 197  df-an 386

This theorem is referenced by:  euind  3393  reuind  3411  reusv3i  4875  opelopabt  4987  wemaplem2  8452  rexanre  14086  rlimcn2  14321  o1of2  14343  o1rlimmul  14349  2sqlem6  25148  spanuni  28403  bj-mo3OLD  32832  isbasisrelowllem1  33203  isbasisrelowllem2  33204  heicant  33444  pm11.71  38597