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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 6045 | . . 3 ⊢ ◡ I = I | |
2 | 1 | breqi 5080 | . 2 ⊢ (𝐴◡ I 𝐵 ↔ 𝐴 I 𝐵) |
3 | reli 5736 | . . 3 ⊢ Rel I | |
4 | 3 | relbrcnv 6015 | . 2 ⊢ (𝐴◡ I 𝐵 ↔ 𝐵 I 𝐴) |
5 | 2, 4 | bitr3i 276 | 1 ⊢ (𝐴 I 𝐵 ↔ 𝐵 I 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 class class class wbr 5074 I cid 5488 ◡ccnv 5588 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-br 5075 df-opab 5137 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 |
This theorem is referenced by: ideq2 36443 |
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