![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 452 | . 2 ⊢ (((𝜑 ∧ ¬ 𝜓) → 𝜒) ↔ (𝜑 → (¬ 𝜓 → 𝜒))) | |
2 | df-or 847 | . . 3 ⊢ ((𝜓 ∨ 𝜒) ↔ (¬ 𝜓 → 𝜒)) | |
3 | 2 | imbi2i 336 | . 2 ⊢ ((𝜑 → (𝜓 ∨ 𝜒)) ↔ (𝜑 → (¬ 𝜓 → 𝜒))) |
4 | 1, 3 | bitr4i 278 | 1 ⊢ (((𝜑 ∧ ¬ 𝜓) → 𝜒) ↔ (𝜑 → (𝜓 ∨ 𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 846 |
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 206 df-an 398 df-or 847 |
This theorem is referenced by: ssundif 4449 brdom3 10472 grothprim 10778 eliccelico 31734 elicoelioo 31735 ballotlemfc0 33156 ballotlemfcc 33157 elicc3 34842 faosnf0.11b 41791 ifpidg 41855 dfsucon 41887 icccncfext 44218 |
Copyright terms: Public domain | W3C validator |