Theorem extid 37682

Description: Property of identity relation, see also extep 37654, extssr 37882 and the comment of df-ssr 37871. (Contributed by Peter Mazsa, 5-Jul-2019.)

Assertion

Ref	Expression
extid	⊢ (𝐴 ∈ 𝑉 → ([𝐴]◡ I = [𝐵]◡ I ↔ 𝐴 = 𝐵))

Proof of Theorem extid

Step	Hyp	Ref	Expression
1		cnvi 6132	. . . . 5 ⊢ ◡ I = I
2	1	eceq2i 8741	. . . 4 ⊢ [𝐴]◡ I = [𝐴] I
3		ecidsn 8753	. . . 4 ⊢ [𝐴] I = {𝐴}
4	2, 3	eqtri 2752	. . 3 ⊢ [𝐴]◡ I = {𝐴}
5	1	eceq2i 8741	. . . 4 ⊢ [𝐵]◡ I = [𝐵] I
6		ecidsn 8753	. . . 4 ⊢ [𝐵] I = {𝐵}
7	5, 6	eqtri 2752	. . 3 ⊢ [𝐵]◡ I = {𝐵}
8	4, 7	eqeq12i 2742	. 2 ⊢ ([𝐴]◡ I = [𝐵]◡ I ↔ {𝐴} = {𝐵})
9		sneqbg 4837	. 2 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} ↔ 𝐴 = 𝐵))
10	8, 9	bitrid 283	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴]◡ I = [𝐵]◡ I ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1533   ∈ wcel 2098  {csn 4621   I cid 5564  ^◡ccnv 5666  [cec 8698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-ec 8702

This theorem is referenced by: (None)