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Mirrors > Home > NFE Home > Th. List > neeqtrd | GIF version |
Description: Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.) |
Ref | Expression |
---|---|
neeqtrd.1 | ⊢ (φ → A ≠ B) |
neeqtrd.2 | ⊢ (φ → B = C) |
Ref | Expression |
---|---|
neeqtrd | ⊢ (φ → A ≠ C) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neeqtrd.1 | . 2 ⊢ (φ → A ≠ B) | |
2 | neeqtrd.2 | . . 3 ⊢ (φ → B = C) | |
3 | 2 | neeq2d 2530 | . 2 ⊢ (φ → (A ≠ B ↔ A ≠ C)) |
4 | 1, 3 | mpbid 201 | 1 ⊢ (φ → A ≠ C) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = 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: neeqtrrd 2540 |
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