Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > necon2bd | GIF version |
Description: Contrapositive inference for inequality. (Contributed by NM, 13-Apr-2007.) |
Ref | Expression |
---|---|
necon2bd.1 | ⊢ (𝜑 → (𝜓 → 𝐴 ≠ 𝐵)) |
Ref | Expression |
---|---|
necon2bd | ⊢ (𝜑 → (𝐴 = 𝐵 → ¬ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | necon2bd.1 | . . 3 ⊢ (𝜑 → (𝜓 → 𝐴 ≠ 𝐵)) | |
2 | df-ne 2246 | . . 3 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵) | |
3 | 1, 2 | syl6ib 159 | . 2 ⊢ (𝜑 → (𝜓 → ¬ 𝐴 = 𝐵)) |
4 | 3 | con2d 586 | 1 ⊢ (𝜑 → (𝐴 = 𝐵 → ¬ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1284 ≠ wne 2245 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-in1 576 ax-in2 577 |
This theorem depends on definitions: df-bi 115 df-ne 2246 |
This theorem is referenced by: nneo 8450 zeo2 8453 bezoutr1 10422 coprm 10523 sqrt2irr 10541 |
Copyright terms: Public domain | W3C validator |