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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 472 | . . . 4 ⊢ ((𝜓 ∧ 𝜒) → 𝜓) | |
2 | 1 | imim2i 16 | . . 3 ⊢ ((𝜑 → (𝜓 ∧ 𝜒)) → (𝜑 → 𝜓)) |
3 | simpr 476 | . . . 4 ⊢ ((𝜓 ∧ 𝜒) → 𝜒) | |
4 | 3 | imim2i 16 | . . 3 ⊢ ((𝜑 → (𝜓 ∧ 𝜒)) → (𝜑 → 𝜒)) |
5 | 2, 4 | jca 553 | . 2 ⊢ ((𝜑 → (𝜓 ∧ 𝜒)) → ((𝜑 → 𝜓) ∧ (𝜑 → 𝜒))) |
6 | pm3.43 902 | . 2 ⊢ (((𝜑 → 𝜓) ∧ (𝜑 → 𝜒)) → (𝜑 → (𝜓 ∧ 𝜒))) | |
7 | 5, 6 | impbii 198 | 1 ⊢ ((𝜑 → (𝜓 ∧ 𝜒)) ↔ ((𝜑 → 𝜓) ∧ (𝜑 → 𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ 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: ordi 904 pm4.76 906 pm5.44 948 2mo2 2538 ssconb 3705 ssin 3797 tfr3 7382 trclfvcotr 13598 isprm2 15233 lgsquad2lem2 24910 ostthlem2 25117 pclclN 34195 ifpbibib 36874 elmapintrab 36901 elinintrab 36902 2reu4a 39838 |
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