![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > prth | Structured version Visualization version GIF version |
Description: Conjoin antecedents and consequents of two premises. This is the closed theorem form of anim12d 588. 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.) |
Ref | Expression |
---|---|
prth | ⊢ (((𝜑 → 𝜓) ∧ (𝜒 → 𝜃)) → ((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜃))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 468 | . 2 ⊢ (((𝜑 → 𝜓) ∧ (𝜒 → 𝜃)) → (𝜑 → 𝜓)) | |
2 | simpr 471 | . 2 ⊢ (((𝜑 → 𝜓) ∧ (𝜒 → 𝜃)) → (𝜒 → 𝜃)) | |
3 | 1, 2 | anim12d 588 | 1 ⊢ (((𝜑 → 𝜓) ∧ (𝜒 → 𝜃)) → ((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜃))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 |
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 383 |
This theorem is referenced by: euind 3543 reuind 3561 reusv3i 5003 opelopabt 5120 wemaplem2 8607 rexanre 14293 rlimcn2 14528 o1of2 14550 o1rlimmul 14556 2sqlem6 25368 spanuni 28737 bj-mo3OLD 33161 isbasisrelowllem1 33533 isbasisrelowllem2 33534 heicant 33770 pm11.71 39116 |
Copyright terms: Public domain | W3C validator |