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Mirrors > Home > ILE Home > Th. List > notbid | GIF version |
Description: Deduction negating both sides of a logical equivalence. (Contributed by NM, 21-May-1994.) (Revised by Mario Carneiro, 31-Jan-2015.) |
Ref | Expression |
---|---|
notbid.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
notbid | ⊢ (𝜑 → (¬ 𝜓 ↔ ¬ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | notbid.1 | . . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
2 | 1 | biimprd 157 | . . 3 ⊢ (𝜑 → (𝜒 → 𝜓)) |
3 | 2 | con3d 601 | . 2 ⊢ (𝜑 → (¬ 𝜓 → ¬ 𝜒)) |
4 | 1 | biimpd 143 | . . 3 ⊢ (𝜑 → (𝜓 → 𝜒)) |
5 | 4 | con3d 601 | . 2 ⊢ (𝜑 → (¬ 𝜒 → ¬ 𝜓)) |
6 | 3, 5 | impbid 128 | 1 ⊢ (𝜑 → (¬ 𝜓 ↔ ¬ 𝜒)) |
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