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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 784. (Contributed by Jim Kingdon, 14-Mar-2020.) |
Ref | Expression |
---|---|
dcne | ⊢ (DECID 𝐴 = 𝐵 ↔ (𝐴 = 𝐵 ∨ 𝐴 ≠ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dc 784 | . 2 ⊢ (DECID 𝐴 = 𝐵 ↔ (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵)) | |
2 | df-ne 2263 | . . 3 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵) | |
3 | 2 | orbi2i 717 | . 2 ⊢ ((𝐴 = 𝐵 ∨ 𝐴 ≠ 𝐵) ↔ (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵)) |
4 | 1, 3 | bitr4i 186 | 1 ⊢ (DECID 𝐴 = 𝐵 ↔ (𝐴 = 𝐵 ∨ 𝐴 ≠ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ↔ wb 104 ∨ wo 667 DECID wdc 783 = wceq 1296 ≠ wne 2262 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 668 |
This theorem depends on definitions: df-bi 116 df-dc 784 df-ne 2263 |
This theorem is referenced by: updjudhf 6850 zdceq 8920 nn0lt2 8926 xlesubadd 9449 qdceq 9807 xrmaxadd 10804 nn0seqcvgd 11450 |
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