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Metamath Proof Explorer
Theorem iba
Description: Introduction of antecedent as conjunct. Theorem *4.73 of
WhiteheadRussell p. 121. (Contributed by NM, 30-Mar-1994)
|
|
Ref |
Expression |
|
Assertion |
iba |
⊢ ( 𝜑 → ( 𝜓 ↔ ( 𝜓 ∧ 𝜑 ) ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
pm3.21 |
⊢ ( 𝜑 → ( 𝜓 → ( 𝜓 ∧ 𝜑 ) ) ) |
| 2 |
|
simpl |
⊢ ( ( 𝜓 ∧ 𝜑 ) → 𝜓 ) |
| 3 |
1 2
|
impbid1 |
⊢ ( 𝜑 → ( 𝜓 ↔ ( 𝜓 ∧ 𝜑 ) ) ) |