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Mirrors > Home > MPE Home > Th. List > Mathboxes > brid | Structured version Visualization version GIF version |
Description: Property of the identity binary relation. (Contributed by Peter Mazsa, 18-Dec-2021.) |
Ref | Expression |
---|---|
brid | ⊢ (𝐴 I 𝐵 ↔ 𝐵 I 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvi 5995 | . . 3 ⊢ ◡ I = I | |
2 | 1 | breqi 5065 | . 2 ⊢ (𝐴◡ I 𝐵 ↔ 𝐴 I 𝐵) |
3 | reli 5693 | . . 3 ⊢ Rel I | |
4 | 3 | relbrcnv 5965 | . 2 ⊢ (𝐴◡ I 𝐵 ↔ 𝐵 I 𝐴) |
5 | 2, 4 | bitr3i 279 | 1 ⊢ (𝐴 I 𝐵 ↔ 𝐵 I 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 class class class wbr 5059 I cid 5454 ◡ccnv 5549 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-br 5060 df-opab 5122 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 |
This theorem is referenced by: ideq2 35559 |
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