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Mirrors > Home > ILE Home > Th. List > necomd | GIF version |
Description: Deduction from commutative law for inequality. (Contributed by NM, 12-Feb-2008.) |
Ref | Expression |
---|---|
necomd.1 | ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
Ref | Expression |
---|---|
necomd | ⊢ (𝜑 → 𝐵 ≠ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | necomd.1 | . 2 ⊢ (𝜑 → 𝐴 ≠ 𝐵) | |
2 | necom 2289 | . 2 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴) | |
3 | 1, 2 | sylib 127 | 1 ⊢ (𝜑 → 𝐵 ≠ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ≠ 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 ax-5 1336 ax-gen 1338 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-cleq 2033 df-ne 2206 |
This theorem is referenced by: difsnb 3506 0nelop 3985 fidifsnen 6331 ltned 7131 lt0ne0 7423 zdceq 8316 zneo 8339 xrlttri3 8718 qdceq 9102 flqltnz 9129 expival 9257 |
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