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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 2839 | . 2 ⊢ (𝜑 → (𝐶 = 𝐷 → ¬ 𝐴 ≠ 𝐵)) |
3 | nne 2827 | . 2 ⊢ (¬ 𝐴 ≠ 𝐵 ↔ 𝐴 = 𝐵) | |
4 | 2, 3 | syl6ib 241 | 1 ⊢ (𝜑 → (𝐶 = 𝐷 → 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1523 ≠ wne 2823 |
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 197 df-ne 2824 |
This theorem is referenced by: oa00 7684 map0g 7939 epfrs 8645 fin23lem24 9182 abs00 14073 oddvds 18012 isabvd 18868 01eq0ring 19320 uvcf1 20179 lindff1 20207 hausnei2 21205 dfconn2 21270 hausflimi 21831 hauspwpwf1 21838 cxpeq0 24469 his6 28084 nn0sqeq1 29641 lkreqN 34775 ltrnideq 35780 hdmapip0 37524 rpnnen3 37916 |
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