Theorem extid 38850

Description: Property of identity relation, see also extep 38823, extssr 39123 and the comment of df-ssr 39112. (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 5869	. . . . 5 ⊢ ◡ I = I
2	1	eceq2i 8733	. . . 4 ⊢ [𝐴]◡ I = [𝐴] I
3		ecidsn 8749	. . . 4 ⊢ [𝐴] I = {𝐴}
4	2, 3	eqtri 2792	. . 3 ⊢ [𝐴]◡ I = {𝐴}
5	1	eceq2i 8733	. . . 4 ⊢ [𝐵]◡ I = [𝐵] I
6		ecidsn 8749	. . . 4 ⊢ [𝐵] I = {𝐵}
7	5, 6	eqtri 2792	. . 3 ⊢ [𝐵]◡ I = {𝐵}
8	4, 7	eqeq12i 2787	. 2 ⊢ ([𝐴]◡ I = [𝐵]◡ I ↔ {𝐴} = {𝐵})
9		sneqbg 4809	. 2 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} ↔ 𝐴 = 𝐵))
10	8, 9	bitrid 286	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴]◡ I = [𝐵]◡ I ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1567   ∈ wcel 2149  {csn 4591   I cid 5553  ^◡ccnv 5658  [cec 8688

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5258  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5175  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ec 8692

This theorem is referenced by: (None)