Theorem extid 38048

Description: Property of identity relation, see also extep 38021, extssr 38247 and the comment of df-ssr 38236. (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 6156	. . . . 5 ⊢ ◡ I = I
2	1	eceq2i 8780	. . . 4 ⊢ [𝐴]◡ I = [𝐴] I
3		ecidsn 8793	. . . 4 ⊢ [𝐴] I = {𝐴}
4	2, 3	eqtri 2757	. . 3 ⊢ [𝐴]◡ I = {𝐴}
5	1	eceq2i 8780	. . . 4 ⊢ [𝐵]◡ I = [𝐵] I
6		ecidsn 8793	. . . 4 ⊢ [𝐵] I = {𝐵}
7	5, 6	eqtri 2757	. . 3 ⊢ [𝐵]◡ I = {𝐵}
8	4, 7	eqeq12i 2747	. 2 ⊢ ([𝐴]◡ I = [𝐵]◡ I ↔ {𝐴} = {𝐵})
9		sneqbg 4853	. 2 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} ↔ 𝐴 = 𝐵))
10	8, 9	bitrid 282	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴]◡ I = [𝐵]◡ I ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1534   ∈ wcel 2100  {csn 4634   I cid 5581  ^◡ccnv 5683  [cec 8736

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2102  ax-9 2110  ax-ext 2700  ax-sep 5305  ax-nul 5312  ax-pr 5435

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2062  df-clab 2707  df-cleq 2721  df-clel 2806  df-ral 3055  df-rex 3064  df-rab 3429  df-v 3474  df-dif 3961  df-un 3963  df-in 3965  df-ss 3975  df-nul 4334  df-if 4535  df-sn 4635  df-pr 4637  df-op 4641  df-br 5155  df-opab 5217  df-id 5582  df-xp 5690  df-rel 5691  df-cnv 5692  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-ec 8740

This theorem is referenced by: (None)