Theorem ideq2 38341

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 38340	. 2 ⊢ (𝐴 I 𝐵 ↔ 𝐵 I 𝐴)
2		ideqg 5786	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐵 I 𝐴 ↔ 𝐵 = 𝐴))
3		eqcom 2738	. . 3 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵)
4	2, 3	bitrdi 287	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐵 I 𝐴 ↔ 𝐴 = 𝐵))
5	1, 4	bitrid 283	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1541   ∈ wcel 2111   class class class wbr 5086   I cid 5505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-br 5087  df-opab 5149  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619

This theorem is referenced by:  br1cossinidres  38486  br1cossxrnidres  38488