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Mirrors > Home > NFE Home > Th. List > neneqad | GIF version |
Description: If it is not the case that two classes are equal, they are unequal. Converse of neneqd 2532. One-way deduction form of df-ne 2518. (Contributed by David Moews, 28-Feb-2017.) |
Ref | Expression |
---|---|
neneqad.1 | ⊢ (φ → ¬ A = B) |
Ref | Expression |
---|---|
neneqad | ⊢ (φ → A ≠ B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neneqad.1 | . . 3 ⊢ (φ → ¬ A = B) | |
2 | 1 | con2i 112 | . 2 ⊢ (A = B → ¬ φ) |
3 | 2 | necon2ai 2561 | 1 ⊢ (φ → A ≠ B) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1642 ≠ wne 2516 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 177 df-ne 2518 |
This theorem is referenced by: (None) |
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