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Mirrors > Home > MPE Home > Th. List > jcab | Structured version Visualization version GIF version |
Description: Distributive law for implication over conjunction. Compare Theorem *4.76 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Apr-1994.) (Proof shortened by Wolf Lammen, 27-Nov-2013.) |
Ref | Expression |
---|---|
jcab | ⊢ ((𝜑 → (𝜓 ∧ 𝜒)) ↔ ((𝜑 → 𝜓) ∧ (𝜑 → 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 483 | . . . 4 ⊢ ((𝜓 ∧ 𝜒) → 𝜓) | |
2 | 1 | imim2i 16 | . . 3 ⊢ ((𝜑 → (𝜓 ∧ 𝜒)) → (𝜑 → 𝜓)) |
3 | simpr 485 | . . . 4 ⊢ ((𝜓 ∧ 𝜒) → 𝜒) | |
4 | 3 | imim2i 16 | . . 3 ⊢ ((𝜑 → (𝜓 ∧ 𝜒)) → (𝜑 → 𝜒)) |
5 | 2, 4 | jca 512 | . 2 ⊢ ((𝜑 → (𝜓 ∧ 𝜒)) → ((𝜑 → 𝜓) ∧ (𝜑 → 𝜒))) |
6 | pm3.43 474 | . 2 ⊢ (((𝜑 → 𝜓) ∧ (𝜑 → 𝜒)) → (𝜑 → (𝜓 ∧ 𝜒))) | |
7 | 5, 6 | impbii 208 | 1 ⊢ ((𝜑 → (𝜓 ∧ 𝜒)) ↔ ((𝜑 → 𝜓) ∧ (𝜑 → 𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 |
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 206 df-an 397 |
This theorem is referenced by: pm4.76 519 pm5.44 543 ordi 1004 2mo2 2643 ssconb 4137 ssin 4230 2reu4lem 4525 tfr3 8398 trclfvcotr 14955 isprm2 16618 lgsquad2lem2 26885 ostthlem2 27128 pclclN 38757 ifpbibib 42251 elmapintrab 42317 elinintrab 42318 ismnuprim 43043 |
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