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Mirrors > Home > ILE Home > Th. List > releq | GIF version |
Description: Equality theorem for the relation predicate. (Contributed by NM, 1-Aug-1994.) |
Ref | Expression |
---|---|
releq | ⊢ (𝐴 = 𝐵 → (Rel 𝐴 ↔ Rel 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseq1 3160 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 ⊆ (V × V) ↔ 𝐵 ⊆ (V × V))) | |
2 | df-rel 4605 | . 2 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V)) | |
3 | df-rel 4605 | . 2 ⊢ (Rel 𝐵 ↔ 𝐵 ⊆ (V × V)) | |
4 | 1, 2, 3 | 3bitr4g 222 | 1 ⊢ (𝐴 = 𝐵 → (Rel 𝐴 ↔ Rel 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1342 Vcvv 2721 ⊆ wss 3111 × cxp 4596 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: releqi 4681 releqd 4682 dfrel2 5048 tposfn2 6225 ereq1 6499 |
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