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Mirrors > Home > ILE Home > Th. List > neeq2d | GIF version |
Description: Deduction for inequality. (Contributed by NM, 25-Oct-1999.) |
Ref | Expression |
---|---|
neeq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
neeq2d | ⊢ (𝜑 → (𝐶 ≠ 𝐴 ↔ 𝐶 ≠ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neeq1d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | neeq2 2361 | . 2 ⊢ (𝐴 = 𝐵 → (𝐶 ≠ 𝐴 ↔ 𝐶 ≠ 𝐵)) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → (𝐶 ≠ 𝐴 ↔ 𝐶 ≠ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1353 ≠ wne 2347 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-5 1447 ax-gen 1449 ax-4 1510 ax-17 1526 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-cleq 2170 df-ne 2348 |
This theorem is referenced by: neeq12d 2367 neeqtrd 2375 sqrt2irr 12154 ennnfonelemk 12393 ennnfoneleminc 12404 ennnfonelemex 12407 ennnfonelemnn0 12415 ennnfonelemr 12416 setscomd 12495 |
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