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Mirrors > Home > NFE Home > Th. List > neneqd | GIF version |
Description: Deduction eliminating inequality definition. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
Ref | Expression |
---|---|
neneqd.1 | ⊢ (φ → A ≠ B) |
Ref | Expression |
---|---|
neneqd | ⊢ (φ → ¬ A = B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neneqd.1 | . 2 ⊢ (φ → A ≠ B) | |
2 | df-ne 2518 | . 2 ⊢ (A ≠ B ↔ ¬ A = B) | |
3 | 1, 2 | sylib 188 | 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: necon2bi 2562 necon2i 2563 pm2.21ddne 2590 nulnnn 4556 enprmaplem3 6078 |
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