Theorem anim12 342

Description: 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		simpl 108	. 2 ⊢ (((𝜑 → 𝜓) ∧ (𝜒 → 𝜃)) → (𝜑 → 𝜓))
2		simpr 109	. 2 ⊢ (((𝜑 → 𝜓) ∧ (𝜒 → 𝜃)) → (𝜒 → 𝜃))
3	1, 2	anim12d 333	1 ⊢ (((𝜑 → 𝜓) ∧ (𝜒 → 𝜃)) → ((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜃)))

Syntax hints:   → wi 4   ∧ wa 103

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  nfand  1561  equsexd  1722  mo23  2060  euind  2917  reuind  2935  reuss2  3407  opelopabt  4247  reusv3i  4444  rexanre  11184  2sqlem6  13750  bj-stan  13782