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Mirrors > Home > MPE Home > Th. List > necon1d | Structured version Visualization version GIF version |
Description: Contrapositive law deduction for inequality. (Contributed by NM, 28-Dec-2008.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Ref | Expression |
---|---|
necon1d.1 | ⊢ (𝜑 → (𝐴 ≠ 𝐵 → 𝐶 = 𝐷)) |
Ref | Expression |
---|---|
necon1d | ⊢ (𝜑 → (𝐶 ≠ 𝐷 → 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | necon1d.1 | . . 3 ⊢ (𝜑 → (𝐴 ≠ 𝐵 → 𝐶 = 𝐷)) | |
2 | nne 2944 | . . 3 ⊢ (¬ 𝐶 ≠ 𝐷 ↔ 𝐶 = 𝐷) | |
3 | 1, 2 | imbitrrdi 251 | . 2 ⊢ (𝜑 → (𝐴 ≠ 𝐵 → ¬ 𝐶 ≠ 𝐷)) |
4 | 3 | necon4ad 2959 | 1 ⊢ (𝜑 → (𝐶 ≠ 𝐷 → 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1541 ≠ wne 2940 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 206 df-ne 2941 |
This theorem is referenced by: disji 5131 mul02lem2 11393 mhpmulcl 21698 xblss2ps 23914 xblss2 23915 lgsne0 26845 h1datomi 30872 eigorthi 31128 disjif 31847 lineintmo 35198 poimirlem6 36580 poimirlem7 36581 2llnmat 38481 2lnat 38741 tendospcanN 39980 dihmeetlem13N 40276 dochkrshp 40343 remul02 41360 remul01 41362 sn-0tie0 41394 |
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