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Mirrors > Home > ILE Home > Th. List > dcne | GIF version |
Description: Decidable equality expressed in terms of ≠. Basically the same as df-dc 743. (Contributed by Jim Kingdon, 14-Mar-2020.) |
Ref | Expression |
---|---|
dcne | ⊢ (DECID 𝐴 = 𝐵 ↔ (𝐴 = 𝐵 ∨ 𝐴 ≠ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dc 743 | . 2 ⊢ (DECID 𝐴 = 𝐵 ↔ (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵)) | |
2 | df-ne 2206 | . . 3 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵) | |
3 | 2 | orbi2i 679 | . 2 ⊢ ((𝐴 = 𝐵 ∨ 𝐴 ≠ 𝐵) ↔ (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵)) |
4 | 1, 3 | bitr4i 176 | 1 ⊢ (DECID 𝐴 = 𝐵 ↔ (𝐴 = 𝐵 ∨ 𝐴 ≠ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ↔ wb 98 ∨ wo 629 DECID wdc 742 = wceq 1243 ≠ wne 2204 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-ne 2206 |
This theorem is referenced by: zdceq 8316 nn0lt2 8322 qdceq 9102 nn0seqcvgd 9880 |
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