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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 2322 | . 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 104 = wceq 1331 ≠ wne 2308 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-5 1423 ax-gen 1425 ax-4 1487 ax-17 1506 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-cleq 2132 df-ne 2309 |
This theorem is referenced by: neeq12d 2328 neeqtrd 2336 sqrt2irr 11840 ennnfonelemk 11913 ennnfoneleminc 11924 ennnfonelemex 11927 ennnfonelemnn0 11935 ennnfonelemr 11936 |
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