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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 584. 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 472 | . 2 ⊢ (((𝜑 → 𝜓) ∧ (𝜒 → 𝜃)) → (𝜑 → 𝜓)) | |
2 | simpr 476 | . 2 ⊢ (((𝜑 → 𝜓) ∧ (𝜒 → 𝜃)) → (𝜒 → 𝜃)) | |
3 | 1, 2 | anim12d 584 | 1 ⊢ (((𝜑 → 𝜓) ∧ (𝜒 → 𝜃)) → ((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜃))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 |
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 196 df-an 385 |
This theorem is referenced by: euind 3360 reuind 3378 reusv3i 4801 opelopabt 4912 wemaplem2 8335 rexanre 13934 rlimcn2 14169 o1of2 14191 o1rlimmul 14197 2sqlem6 24948 spanuni 27787 bj-mo3OLD 32022 isbasisrelowllem1 32379 isbasisrelowllem2 32380 heicant 32614 pm11.71 37619 |
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