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Mirrors > Home > ILE Home > Th. List > releqd | GIF version |
Description: Equality deduction for the relation predicate. (Contributed by NM, 8-Mar-2014.) |
Ref | Expression |
---|---|
releqd.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
releqd | ⊢ (𝜑 → (Rel 𝐴 ↔ Rel 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | releqd.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | releq 4680 | . 2 ⊢ (𝐴 = 𝐵 → (Rel 𝐴 ↔ Rel 𝐵)) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → (Rel 𝐴 ↔ Rel 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1342 Rel wrel 4603 |
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-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-11 1493 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-in 3117 df-ss 3124 df-rel 4605 |
This theorem is referenced by: dftpos3 6221 tposfo2 6226 tposf12 6228 lmreltop 12740 cnprcl2k 12753 |
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