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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 453 | . 2 ⊢ (((𝜑 ∧ ¬ 𝜓) → 𝜒) ↔ (𝜑 → (¬ 𝜓 → 𝜒))) | |
2 | df-or 844 | . . 3 ⊢ ((𝜓 ∨ 𝜒) ↔ (¬ 𝜓 → 𝜒)) | |
3 | 2 | imbi2i 338 | . 2 ⊢ ((𝜑 → (𝜓 ∨ 𝜒)) ↔ (𝜑 → (¬ 𝜓 → 𝜒))) |
4 | 1, 3 | bitr4i 280 | 1 ⊢ (((𝜑 ∧ ¬ 𝜓) → 𝜒) ↔ (𝜑 → (𝜓 ∨ 𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 |
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 209 df-an 399 df-or 844 |
This theorem is referenced by: ssundif 4432 brdom3 9949 grothprim 10255 eliccelico 30499 elicoelioo 30500 ballotlemfc0 31750 ballotlemfcc 31751 elicc3 33665 ifpidg 39855 dfsucon 39887 icccncfext 42168 |
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