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Mirrors > Home > ILE Home > Th. List > nnedc | GIF version |
Description: Negation of inequality where equality is decidable. (Contributed by Jim Kingdon, 15-May-2018.) |
Ref | Expression |
---|---|
nnedc | ⊢ (DECID 𝐴 = 𝐵 → (¬ 𝐴 ≠ 𝐵 ↔ 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ne 2328 | . . . 4 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵) | |
2 | 1 | a1i 9 | . . 3 ⊢ (DECID 𝐴 = 𝐵 → (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵)) |
3 | 2 | con2biidc 865 | . 2 ⊢ (DECID 𝐴 = 𝐵 → (𝐴 = 𝐵 ↔ ¬ 𝐴 ≠ 𝐵)) |
4 | 3 | bicomd 140 | 1 ⊢ (DECID 𝐴 = 𝐵 → (¬ 𝐴 ≠ 𝐵 ↔ 𝐴 = 𝐵)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 104 DECID wdc 820 = wceq 1335 ≠ wne 2327 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-ne 2328 |
This theorem is referenced by: nn0n0n1ge2b 9248 alzdvds 11758 fzo0dvdseq 11761 algcvgblem 11941 |
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