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Mirrors > Home > ILE Home > Th. List > necon3bbii | GIF version |
Description: Deduction from equality to inequality. (Contributed by NM, 13-Apr-2007.) |
Ref | Expression |
---|---|
necon3bbii.1 | ⊢ (𝜑 ↔ 𝐴 = 𝐵) |
Ref | Expression |
---|---|
necon3bbii | ⊢ (¬ 𝜑 ↔ 𝐴 ≠ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | necon3bbii.1 | . . . 4 ⊢ (𝜑 ↔ 𝐴 = 𝐵) | |
2 | 1 | bicomi 130 | . . 3 ⊢ (𝐴 = 𝐵 ↔ 𝜑) |
3 | 2 | necon3abii 2281 | . 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝜑) |
4 | 3 | bicomi 130 | 1 ⊢ (¬ 𝜑 ↔ 𝐴 ≠ 𝐵) |
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: (None) |
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