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Mirrors > Home > ILE Home > Th. List > necon2ad | GIF version |
Description: Contrapositive inference for inequality. (Contributed by NM, 19-Apr-2007.) (Proof rewritten by Jim Kingdon, 16-May-2018.) |
Ref | Expression |
---|---|
necon2ad.1 | ⊢ (𝜑 → (𝐴 = 𝐵 → ¬ 𝜓)) |
Ref | Expression |
---|---|
necon2ad | ⊢ (𝜑 → (𝜓 → 𝐴 ≠ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | necon2ad.1 | . . 3 ⊢ (𝜑 → (𝐴 = 𝐵 → ¬ 𝜓)) | |
2 | 1 | con2d 554 | . 2 ⊢ (𝜑 → (𝜓 → ¬ 𝐴 = 𝐵)) |
3 | df-ne 2206 | . 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵) | |
4 | 2, 3 | syl6ibr 151 | 1 ⊢ (𝜑 → (𝜓 → 𝐴 ≠ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1243 ≠ wne 2204 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 |
This theorem depends on definitions: df-bi 110 df-ne 2206 |
This theorem is referenced by: necon2d 2264 prneimg 3545 tz7.2 4091 nordeq 4268 ltne 7103 apne 7614 xrltne 8729 ge0nemnf 8737 sqrt2irr 9878 |
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