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Metamath Proof Explorer
Description: Theorem *4.43 of WhiteheadRussell p. 119. (Contributed by NM, 3-Jan-2005) (Proof shortened by Wolf Lammen, 26-Nov-2012)
|
|
Ref |
Expression |
|
Assertion |
pm4.43 |
⊢ ( 𝜑 ↔ ( ( 𝜑 ∨ 𝜓 ) ∧ ( 𝜑 ∨ ¬ 𝜓 ) ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
pm3.24 |
⊢ ¬ ( 𝜓 ∧ ¬ 𝜓 ) |
| 2 |
1
|
biorfri |
⊢ ( 𝜑 ↔ ( 𝜑 ∨ ( 𝜓 ∧ ¬ 𝜓 ) ) ) |
| 3 |
|
ordi |
⊢ ( ( 𝜑 ∨ ( 𝜓 ∧ ¬ 𝜓 ) ) ↔ ( ( 𝜑 ∨ 𝜓 ) ∧ ( 𝜑 ∨ ¬ 𝜓 ) ) ) |
| 4 |
2 3
|
bitri |
⊢ ( 𝜑 ↔ ( ( 𝜑 ∨ 𝜓 ) ∧ ( 𝜑 ∨ ¬ 𝜓 ) ) ) |