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Metamath Proof Explorer
Description: Theorem *5.61 of WhiteheadRussell p. 125. (Contributed by NM, 3-Jan-2005) (Proof shortened by Wolf Lammen, 30-Jun-2013)
|
|
Ref |
Expression |
|
Assertion |
pm5.61 |
⊢ ( ( ( 𝜑 ∨ 𝜓 ) ∧ ¬ 𝜓 ) ↔ ( 𝜑 ∧ ¬ 𝜓 ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
orel2 |
⊢ ( ¬ 𝜓 → ( ( 𝜑 ∨ 𝜓 ) → 𝜑 ) ) |
| 2 |
|
orc |
⊢ ( 𝜑 → ( 𝜑 ∨ 𝜓 ) ) |
| 3 |
1 2
|
impbid1 |
⊢ ( ¬ 𝜓 → ( ( 𝜑 ∨ 𝜓 ) ↔ 𝜑 ) ) |
| 4 |
3
|
pm5.32ri |
⊢ ( ( ( 𝜑 ∨ 𝜓 ) ∧ ¬ 𝜓 ) ↔ ( 𝜑 ∧ ¬ 𝜓 ) ) |