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Metamath Proof Explorer
Description: Deduction negating both sides of a logical equivalence. (Contributed by NM, 21-May-1994)
|
|
Ref |
Expression |
|
Hypothesis |
notbid.1 |
⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) |
|
Assertion |
notbid |
⊢ ( 𝜑 → ( ¬ 𝜓 ↔ ¬ 𝜒 ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
notbid.1 |
⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) ) |
| 2 |
|
notnotb |
⊢ ( 𝜓 ↔ ¬ ¬ 𝜓 ) |
| 3 |
|
notnotb |
⊢ ( 𝜒 ↔ ¬ ¬ 𝜒 ) |
| 4 |
1 2 3
|
3bitr3g |
⊢ ( 𝜑 → ( ¬ ¬ 𝜓 ↔ ¬ ¬ 𝜒 ) ) |
| 5 |
4
|
con4bid |
⊢ ( 𝜑 → ( ¬ 𝜓 ↔ ¬ 𝜒 ) ) |