Theorem brid 38981

Description: Property of the identity binary relation. (Contributed by Peter Mazsa, 18-Dec-2021.)

Assertion

Ref	Expression
brid	⊢ (𝐴 I 𝐵 ↔ 𝐵 I 𝐴)

Proof of Theorem brid

Step	Hyp	Ref	Expression
1		cnvi 5871	. . 3 ⊢ ◡ I = I
2	1	breqi 5115	. 2 ⊢ (𝐴◡ I 𝐵 ↔ 𝐴 I 𝐵)
3		reli 5813	. . 3 ⊢ Rel I
4	3	relbrcnv 6109	. 2 ⊢ (𝐴◡ I 𝐵 ↔ 𝐵 I 𝐴)
5	2, 4	bitr3i 280	1 ⊢ (𝐴 I 𝐵 ↔ 𝐵 I 𝐴)

Syntax hints:   ↔ wb 209   class class class wbr 5109   I cid 5555  ^◡ccnv 5660

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669

This theorem is referenced by:  ideq2  38982