Theorem ideq2 37176

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 37175	. 2 ⊢ (𝐴 I 𝐵 ↔ 𝐵 I 𝐴)
2		ideqg 5852	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐵 I 𝐴 ↔ 𝐵 = 𝐴))
3		eqcom 2740	. . 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 205   = wceq 1542   ∈ wcel 2107   class class class wbr 5149   I cid 5574

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685

This theorem is referenced by:  br1cossinidres  37319  br1cossxrnidres  37321