Theorem extid 35012

Description: Property of identity relation, see also extep 34984, extssr 35194 and the comment of df-ssr 35183. (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 5834	. . . . 5 ⊢ ◡ I = I
2	1	eceq2i 8124	. . . 4 ⊢ [𝐴]◡ I = [𝐴] I
3		ecidsn 8136	. . . 4 ⊢ [𝐴] I = {𝐴}
4	2, 3	eqtri 2796	. . 3 ⊢ [𝐴]◡ I = {𝐴}
5	1	eceq2i 8124	. . . 4 ⊢ [𝐵]◡ I = [𝐵] I
6		ecidsn 8136	. . . 4 ⊢ [𝐵] I = {𝐵}
7	5, 6	eqtri 2796	. . 3 ⊢ [𝐵]◡ I = {𝐵}
8	4, 7	eqeq12i 2786	. 2 ⊢ ([𝐴]◡ I = [𝐵]◡ I ↔ {𝐴} = {𝐵})
9		sneqbg 4642	. 2 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} ↔ 𝐴 = 𝐵))
10	8, 9	syl5bb 275	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴]◡ I = [𝐵]◡ I ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 198   = wceq 1507   ∈ wcel 2050  {csn 4435   I cid 5305  ^◡ccnv 5400  [cec 8081

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-sep 5054  ax-nul 5061  ax-pr 5180

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-br 4924  df-opab 4986  df-id 5306  df-xp 5407  df-rel 5408  df-cnv 5409  df-dm 5411  df-rn 5412  df-res 5413  df-ima 5414  df-ec 8085

This theorem is referenced by: (None)