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Mirrors > Home > MPE Home > Th. List > necon4d | Structured version Visualization version GIF version |
Description: Contrapositive inference for inequality. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Ref | Expression |
---|---|
necon4d.1 | ⊢ (𝜑 → (𝐴 ≠ 𝐵 → 𝐶 ≠ 𝐷)) |
Ref | Expression |
---|---|
necon4d | ⊢ (𝜑 → (𝐶 = 𝐷 → 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | necon4d.1 | . . 3 ⊢ (𝜑 → (𝐴 ≠ 𝐵 → 𝐶 ≠ 𝐷)) | |
2 | 1 | necon2bd 2954 | . 2 ⊢ (𝜑 → (𝐶 = 𝐷 → ¬ 𝐴 ≠ 𝐵)) |
3 | nne 2942 | . 2 ⊢ (¬ 𝐴 ≠ 𝐵 ↔ 𝐴 = 𝐵) | |
4 | 2, 3 | imbitrdi 251 | 1 ⊢ (𝜑 → (𝐶 = 𝐷 → 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1537 ≠ wne 2938 |
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 207 df-ne 2939 |
This theorem is referenced by: oa00 8596 map0g 8923 epfrs 9769 fin23lem24 10360 abs00 15325 oddvds 19580 01eq0ringOLD 20548 isdomn4 20733 isabvd 20830 uvcf1 21830 lindff1 21858 hausnei2 23377 dfconn2 23443 hausflimi 24004 hauspwpwf1 24011 cxpeq0 26735 his6 31128 fnpreimac 32688 deg1le0eq0 33578 lkreqN 39152 ltrnideq 40158 hdmapip0 41898 sticksstones2 42129 unitscyglem4 42180 rpnnen3 43021 |
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