Theorem extid 38328

Description: Property of identity relation, see also extep 38301, extssr 38527 and the comment of df-ssr 38516. (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 6130	. . . . 5 ⊢ ◡ I = I
2	1	eceq2i 8761	. . . 4 ⊢ [𝐴]◡ I = [𝐴] I
3		ecidsn 8774	. . . 4 ⊢ [𝐴] I = {𝐴}
4	2, 3	eqtri 2758	. . 3 ⊢ [𝐴]◡ I = {𝐴}
5	1	eceq2i 8761	. . . 4 ⊢ [𝐵]◡ I = [𝐵] I
6		ecidsn 8774	. . . 4 ⊢ [𝐵] I = {𝐵}
7	5, 6	eqtri 2758	. . 3 ⊢ [𝐵]◡ I = {𝐵}
8	4, 7	eqeq12i 2753	. 2 ⊢ ([𝐴]◡ I = [𝐵]◡ I ↔ {𝐴} = {𝐵})
9		sneqbg 4819	. 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 4601   I cid 5547  ^◡ccnv 5653  [cec 8717

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 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-br 5120  df-opab 5182  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-ec 8721

This theorem is referenced by: (None)