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Mirrors > Home > NFE Home > Th. List > neeq2 | GIF version |
Description: Equality theorem for inequality. (Contributed by NM, 19-Nov-1994.) |
Ref | Expression |
---|---|
neeq2 | ⊢ (A = B → (C ≠ A ↔ C ≠ B)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq2 2362 | . . 3 ⊢ (A = B → (C = A ↔ C = B)) | |
2 | 1 | notbid 285 | . 2 ⊢ (A = B → (¬ C = A ↔ ¬ C = B)) |
3 | df-ne 2518 | . 2 ⊢ (C ≠ A ↔ ¬ C = A) | |
4 | df-ne 2518 | . 2 ⊢ (C ≠ B ↔ ¬ C = B) | |
5 | 2, 3, 4 | 3bitr4g 279 | 1 ⊢ (A = B → (C ≠ A ↔ C ≠ B)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 176 = wceq 1642 ≠ wne 2516 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-11 1746 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-ex 1542 df-cleq 2346 df-ne 2518 |
This theorem is referenced by: neeq2i 2527 neeq2d 2530 psseq2 3357 nfunv 5138 enprmapc 6083 ce2 6192 |
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