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Mirrors > Home > ILE Home > Th. List > dcned | GIF version |
Description: Decidable equality implies decidable negated equality. (Contributed by Jim Kingdon, 3-May-2020.) |
Ref | Expression |
---|---|
dcned.eq | ⊢ (𝜑 → DECID 𝐴 = 𝐵) |
Ref | Expression |
---|---|
dcned | ⊢ (𝜑 → DECID 𝐴 ≠ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dcned.eq | . . 3 ⊢ (𝜑 → DECID 𝐴 = 𝐵) | |
2 | dcn 812 | . . 3 ⊢ (DECID 𝐴 = 𝐵 → DECID ¬ 𝐴 = 𝐵) | |
3 | 1, 2 | syl 14 | . 2 ⊢ (𝜑 → DECID ¬ 𝐴 = 𝐵) |
4 | df-ne 2286 | . . 3 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵) | |
5 | 4 | dcbii 810 | . 2 ⊢ (DECID 𝐴 ≠ 𝐵 ↔ DECID ¬ 𝐴 = 𝐵) |
6 | 3, 5 | sylibr 133 | 1 ⊢ (𝜑 → DECID 𝐴 ≠ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 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-in1 588 ax-in2 589 ax-io 683 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-ne 2286 |
This theorem is referenced by: nn0n0n1ge2b 9098 algcvgblem 11657 |
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