Theorem ideq2 36422

Description: For sets, the identity binary relation is the same as equality. (Contributed by Peter Mazsa, 24-Jun-2020.) (Revised by Peter Mazsa, 18-Dec-2021.)

Assertion

Ref	Expression
ideq2	⊢ (𝐴 ∈ 𝑉 → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))

Proof of Theorem ideq2

Step	Hyp	Ref	Expression
1		brid 36421	. 2 ⊢ (𝐴 I 𝐵 ↔ 𝐵 I 𝐴)
2		ideqg 5757	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐵 I 𝐴 ↔ 𝐵 = 𝐴))
3		eqcom 2746	. . 3 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵)
4	2, 3	bitrdi 286	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐵 I 𝐴 ↔ 𝐴 = 𝐵))
5	1, 4	syl5bb 282	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1541   ∈ wcel 2109   class class class wbr 5078   I cid 5487

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pr 5355

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-br 5079  df-opab 5141  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596

This theorem is referenced by:  br1cossinidres  36546  br1cossxrnidres  36548