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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 210 | 1 ⊢ ((𝜑 → (𝜓 ∧ 𝜒)) ↔ ((𝜑 → 𝜓) ∧ (𝜑 → 𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ 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 208 df-an 397 |
This theorem is referenced by: pm4.76 519 pm5.44 543 ordi 999 2mo2 2725 ssconb 4111 ssin 4204 2reu4lem 4461 tfr3 8024 trclfvcotr 14357 isprm2 16014 lgsquad2lem2 25888 ostthlem2 26131 pclclN 36907 ifpbibib 39754 elmapintrab 39814 elinintrab 39815 ismnuprim 40507 |
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