Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > necon3abii | GIF version |
Description: Deduction from equality to inequality. (Contributed by NM, 9-Nov-2007.) |
Ref | Expression |
---|---|
necon3abii.1 | ⊢ (𝐴 = 𝐵 ↔ 𝜑) |
Ref | Expression |
---|---|
necon3abii | ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ne 2246 | . 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵) | |
2 | necon3abii.1 | . 2 ⊢ (𝐴 = 𝐵 ↔ 𝜑) | |
3 | 1, 2 | xchbinx 639 | 1 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝜑) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ↔ wb 103 = wceq 1284 ≠ wne 2245 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 |
This theorem depends on definitions: df-bi 115 df-ne 2246 |
This theorem is referenced by: necon3bbii 2282 necon3bii 2283 nesym 2290 n0rf 3260 gcd0id 10370 |
Copyright terms: Public domain | W3C validator |