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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 619 | . 2 ⊢ (𝜑 → (𝜓 → ¬ 𝐴 = 𝐵)) |
3 | df-ne 2341 | . 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵) | |
4 | 2, 3 | syl6ibr 161 | 1 ⊢ (𝜑 → (𝜓 → 𝐴 ≠ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1348 ≠ wne 2340 |
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 609 ax-in2 610 |
This theorem depends on definitions: df-bi 116 df-ne 2341 |
This theorem is referenced by: necon2d 2399 prneimg 3761 tz7.2 4339 nordeq 4528 pr2ne 7169 ltne 8004 apne 8542 xrltne 9770 npnflt 9772 nmnfgt 9775 ge0nemnf 9781 rpexp 12107 sqrt2irr 12116 pcgcd1 12281 lgsmod 13721 |
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