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| Mirrors > Home > MPE Home > Th. List > jaob | Structured version Visualization version GIF version | ||
| Description: Disjunction of antecedents. Compare Theorem *4.77 of [WhiteheadRussell] p. 121. (Contributed by NM, 30-May-1994.) (Proof shortened by Wolf Lammen, 9-Dec-2012.) |
| Ref | Expression |
|---|---|
| jaob | ⊢ (((𝜑 ∨ 𝜒) → 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜒 → 𝜓))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | pm2.67-2 892 | . . 3 ⊢ (((𝜑 ∨ 𝜒) → 𝜓) → (𝜑 → 𝜓)) | |
| 2 | olc 869 | . . . 4 ⊢ (𝜒 → (𝜑 ∨ 𝜒)) | |
| 3 | 2 | imim1i 63 | . . 3 ⊢ (((𝜑 ∨ 𝜒) → 𝜓) → (𝜒 → 𝜓)) |
| 4 | 1, 3 | jca 511 | . 2 ⊢ (((𝜑 ∨ 𝜒) → 𝜓) → ((𝜑 → 𝜓) ∧ (𝜒 → 𝜓))) |
| 5 | pm3.44 962 | . 2 ⊢ (((𝜑 → 𝜓) ∧ (𝜒 → 𝜓)) → ((𝜑 ∨ 𝜒) → 𝜓)) | |
| 6 | 4, 5 | impbii 209 | 1 ⊢ (((𝜑 ∨ 𝜒) → 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜒 → 𝜓))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 |
| 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 207 df-an 396 df-or 849 |
| This theorem is referenced by: pm4.77 965 pm5.53 1007 pm4.83 1027 axio 2698 elunant 4184 intprg 4981 relop 5861 sqrt2irr 16285 algcvgblem 16614 efgred 19766 caucfil 25317 plydivex 26339 2sqlem6 27467 arg-ax 36417 tendoeq2 40776 ifpidg 43504 |
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