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Mirrors > Home > NFE Home > Th. List > 3netr4d | GIF version |
Description: Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012.) |
Ref | Expression |
---|---|
3netr4d.1 | ⊢ (φ → A ≠ B) |
3netr4d.2 | ⊢ (φ → C = A) |
3netr4d.3 | ⊢ (φ → D = B) |
Ref | Expression |
---|---|
3netr4d | ⊢ (φ → C ≠ D) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3netr4d.1 | . 2 ⊢ (φ → A ≠ B) | |
2 | 3netr4d.2 | . . 3 ⊢ (φ → C = A) | |
3 | 3netr4d.3 | . . 3 ⊢ (φ → D = B) | |
4 | 2, 3 | neeq12d 2531 | . 2 ⊢ (φ → (C ≠ D ↔ A ≠ B)) |
5 | 1, 4 | mpbird 223 | 1 ⊢ (φ → C ≠ D) |
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: (None) |
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