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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 805. (Contributed by Jim Kingdon, 14-Mar-2020.) |
Ref | Expression |
---|---|
dcne | ⊢ (DECID 𝐴 = 𝐵 ↔ (𝐴 = 𝐵 ∨ 𝐴 ≠ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dc 805 | . 2 ⊢ (DECID 𝐴 = 𝐵 ↔ (𝐴 = 𝐵 ∨ ¬ 𝐴 = 𝐵)) | |
2 | df-ne 2286 | . . 3 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵) | |
3 | 2 | orbi2i 736 | . 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 682 DECID wdc 804 = wceq 1316 ≠ wne 2285 |
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-io 683 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-ne 2286 |
This theorem is referenced by: updjudhf 6932 zdceq 9094 nn0lt2 9100 xlesubadd 9634 qdceq 9992 xrmaxadd 10998 nn0seqcvgd 11649 |
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