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Mirrors > Home > MPE Home > Th. List > releqi | Structured version Visualization version GIF version |
Description: Equality inference for the relation predicate. (Contributed by NM, 8-Dec-2006.) |
Ref | Expression |
---|---|
releqi.1 | ⊢ 𝐴 = 𝐵 |
Ref | Expression |
---|---|
releqi | ⊢ (Rel 𝐴 ↔ Rel 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | releqi.1 | . 2 ⊢ 𝐴 = 𝐵 | |
2 | releq 5124 | . 2 ⊢ (𝐴 = 𝐵 → (Rel 𝐴 ↔ Rel 𝐵)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (Rel 𝐴 ↔ Rel 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 Rel wrel 5043 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-in 3547 df-ss 3554 df-rel 5045 |
This theorem is referenced by: reliun 5162 reluni 5164 relint 5165 reldmmpt2 6669 wfrrel 7307 tfrlem6 7365 relsdom 7848 cda1dif 8881 0rest 15913 firest 15916 2oppchomf 16207 oppchofcl 16723 oyoncl 16733 releqg 17464 reldvdsr 18467 restbas 20772 hlimcaui 27477 relbigcup 31174 fnsingle 31196 funimage 31205 colinrel 31334 neicvgnvor 37434 |
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