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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 746 | . . 3 ⊢ (DECID 𝐴 = 𝐵 → DECID ¬ 𝐴 = 𝐵) | |
3 | 1, 2 | syl 14 | . 2 ⊢ (𝜑 → DECID ¬ 𝐴 = 𝐵) |
4 | df-ne 2206 | . . 3 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵) | |
5 | 4 | dcbii 747 | . 2 ⊢ (DECID 𝐴 ≠ 𝐵 ↔ DECID ¬ 𝐴 = 𝐵) |
6 | 3, 5 | sylibr 137 | 1 ⊢ (𝜑 → DECID 𝐴 ≠ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 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-in1 544 ax-in2 545 ax-io 630 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-ne 2206 |
This theorem is referenced by: nn0n0n1ge2b 8320 algcvgblem 9888 |
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