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Mirrors > Home > NFE Home > Th. List > neeq1d | GIF version |
Description: Deduction for inequality. (Contributed by NM, 25-Oct-1999.) |
Ref | Expression |
---|---|
neeq1d.1 | ⊢ (φ → A = B) |
Ref | Expression |
---|---|
neeq1d | ⊢ (φ → (A ≠ C ↔ B ≠ C)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neeq1d.1 | . 2 ⊢ (φ → A = B) | |
2 | neeq1 2524 | . 2 ⊢ (A = B → (A ≠ C ↔ B ≠ C)) | |
3 | 1, 2 | syl 15 | 1 ⊢ (φ → (A ≠ C ↔ B ≠ C)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 = wceq 1642 ≠ wne 2516 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 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: neeq12d 2531 eqnetrd 2534 prnzg 3836 preaddccan2lem1 4454 preaddccan2 4455 evenodddisj 4516 vfinncvntnn 4548 ereldm 5971 map0 6025 ce0addcnnul 6179 ce0nn 6180 ce0nnulb 6182 |
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