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Metamath Proof Explorer
Description: Theorem *5.63 of WhiteheadRussell p. 125. (Contributed by NM, 3-Jan-2005) (Proof shortened by Wolf Lammen, 25-Dec-2012)
|
|
Ref |
Expression |
|
Assertion |
pm5.63 |
⊢ ( ( 𝜑 ∨ 𝜓 ) ↔ ( 𝜑 ∨ ( ¬ 𝜑 ∧ 𝜓 ) ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
exmid |
⊢ ( 𝜑 ∨ ¬ 𝜑 ) |
| 2 |
|
ordi |
⊢ ( ( 𝜑 ∨ ( ¬ 𝜑 ∧ 𝜓 ) ) ↔ ( ( 𝜑 ∨ ¬ 𝜑 ) ∧ ( 𝜑 ∨ 𝜓 ) ) ) |
| 3 |
1 2
|
mpbiran |
⊢ ( ( 𝜑 ∨ ( ¬ 𝜑 ∧ 𝜓 ) ) ↔ ( 𝜑 ∨ 𝜓 ) ) |
| 4 |
3
|
bicomi |
⊢ ( ( 𝜑 ∨ 𝜓 ) ↔ ( 𝜑 ∨ ( ¬ 𝜑 ∧ 𝜓 ) ) ) |