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Mirrors > Home > ILE Home > Th. List > necon3abid | GIF version |
Description: Deduction from equality to inequality. (Contributed by NM, 21-Mar-2007.) |
Ref | Expression |
---|---|
necon3abid.1 | ⊢ (𝜑 → (𝐴 = 𝐵 ↔ 𝜓)) |
Ref | Expression |
---|---|
necon3abid | ⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ ¬ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ne 2335 | . 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵) | |
2 | necon3abid.1 | . . 3 ⊢ (𝜑 → (𝐴 = 𝐵 ↔ 𝜓)) | |
3 | 2 | notbid 657 | . 2 ⊢ (𝜑 → (¬ 𝐴 = 𝐵 ↔ ¬ 𝜓)) |
4 | 1, 3 | syl5bb 191 | 1 ⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ ¬ 𝜓)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 104 = wceq 1342 ≠ wne 2334 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 |
This theorem depends on definitions: df-bi 116 df-ne 2335 |
This theorem is referenced by: necon3bbid 2374 fndmdif 5587 expnegap0 10457 gcdn0gt0 11905 cncongr2 12030 |
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