Theorem extid 38311

Description: Property of identity relation, see also extep 38284, extssr 38510 and the comment of df-ssr 38499. (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 6161	. . . . 5 ⊢ ◡ I = I
2	1	eceq2i 8787	. . . 4 ⊢ [𝐴]◡ I = [𝐴] I
3		ecidsn 8800	. . . 4 ⊢ [𝐴] I = {𝐴}
4	2, 3	eqtri 2765	. . 3 ⊢ [𝐴]◡ I = {𝐴}
5	1	eceq2i 8787	. . . 4 ⊢ [𝐵]◡ I = [𝐵] I
6		ecidsn 8800	. . . 4 ⊢ [𝐵] I = {𝐵}
7	5, 6	eqtri 2765	. . 3 ⊢ [𝐵]◡ I = {𝐵}
8	4, 7	eqeq12i 2755	. 2 ⊢ ([𝐴]◡ I = [𝐵]◡ I ↔ {𝐴} = {𝐵})
9		sneqbg 4843	. 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 206   = wceq 1540   ∈ wcel 2108  {csn 4626   I cid 5577  ^◡ccnv 5684  [cec 8743

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ec 8747

This theorem is referenced by: (None)