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Mirrors > Home > ILE Home > Th. List > anim12 | GIF version |
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.) |
Ref | Expression |
---|---|
anim12 | ⊢ (((𝜑 → 𝜓) ∧ (𝜒 → 𝜃)) → ((𝜑 ∧ 𝜒) → (𝜓 ∧ 𝜃))) |
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))) |
Colors of variables: wff set class |
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 4245 reusv3i 4442 rexanre 11171 2sqlem6 13671 bj-stan 13703 |
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