Theorem extid 36132

Description: Property of identity relation, see also extep 36104, extssr 36313 and the comment of df-ssr 36302. (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 5985	. . . . 5 ⊢ ◡ I = I
2	1	eceq2i 8410	. . . 4 ⊢ [𝐴]◡ I = [𝐴] I
3		ecidsn 8422	. . . 4 ⊢ [𝐴] I = {𝐴}
4	2, 3	eqtri 2759	. . 3 ⊢ [𝐴]◡ I = {𝐴}
5	1	eceq2i 8410	. . . 4 ⊢ [𝐵]◡ I = [𝐵] I
6		ecidsn 8422	. . . 4 ⊢ [𝐵] I = {𝐵}
7	5, 6	eqtri 2759	. . 3 ⊢ [𝐵]◡ I = {𝐵}
8	4, 7	eqeq12i 2751	. 2 ⊢ ([𝐴]◡ I = [𝐵]◡ I ↔ {𝐴} = {𝐵})
9		sneqbg 4740	. 2 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} ↔ 𝐴 = 𝐵))
10	8, 9	syl5bb 286	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴]◡ I = [𝐵]◡ I ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1543   ∈ wcel 2112  {csn 4527   I cid 5439  ^◡ccnv 5535  [cec 8367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pr 5307

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-br 5040  df-opab 5102  df-id 5440  df-xp 5542  df-rel 5543  df-cnv 5544  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-ec 8371

This theorem is referenced by: (None)