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Mirrors > Home > MPE Home > Th. List > pm5.6 | Structured version Visualization version GIF version |
Description: Conjunction in antecedent versus disjunction in consequent. Theorem *5.6 of [WhiteheadRussell] p. 125. (Contributed by NM, 8-Jun-1994.) |
Ref | Expression |
---|---|
pm5.6 | ⊢ (((𝜑 ∧ ¬ 𝜓) → 𝜒) ↔ (𝜑 → (𝜓 ∨ 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | impexp 454 | . 2 ⊢ (((𝜑 ∧ ¬ 𝜓) → 𝜒) ↔ (𝜑 → (¬ 𝜓 → 𝜒))) | |
2 | df-or 845 | . . 3 ⊢ ((𝜓 ∨ 𝜒) ↔ (¬ 𝜓 → 𝜒)) | |
3 | 2 | imbi2i 339 | . 2 ⊢ ((𝜑 → (𝜓 ∨ 𝜒)) ↔ (𝜑 → (¬ 𝜓 → 𝜒))) |
4 | 1, 3 | bitr4i 281 | 1 ⊢ (((𝜑 ∧ ¬ 𝜓) → 𝜒) ↔ (𝜑 → (𝜓 ∨ 𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 844 |
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 210 df-an 400 df-or 845 |
This theorem is referenced by: ssundif 4391 brdom3 9939 grothprim 10245 eliccelico 30526 elicoelioo 30527 ballotlemfc0 31860 ballotlemfcc 31861 elicc3 33778 ifpidg 40199 dfsucon 40231 icccncfext 42529 |
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