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Mirrors > Home > ILE Home > Th. List > neeq1 | GIF version |
Description: Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.) |
Ref | Expression |
---|---|
neeq1 | ⊢ (𝐴 = 𝐵 → (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2046 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐶)) | |
2 | 1 | notbid 592 | . 2 ⊢ (𝐴 = 𝐵 → (¬ 𝐴 = 𝐶 ↔ ¬ 𝐵 = 𝐶)) |
3 | df-ne 2206 | . 2 ⊢ (𝐴 ≠ 𝐶 ↔ ¬ 𝐴 = 𝐶) | |
4 | df-ne 2206 | . 2 ⊢ (𝐵 ≠ 𝐶 ↔ ¬ 𝐵 = 𝐶) | |
5 | 2, 3, 4 | 3bitr4g 212 | 1 ⊢ (𝐴 = 𝐵 → (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 98 = wceq 1243 ≠ 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-4 1400 ax-17 1419 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-cleq 2033 df-ne 2206 |
This theorem is referenced by: neeq1i 2220 neeq1d 2223 nelrdva 2746 psseq1 3031 0inp0 3919 uzn0 8488 xrnemnf 8699 xrnepnf 8700 ngtmnft 8731 fztpval 8945 |
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