Theorem extid 38505

Description: Property of identity relation, see also extep 38478, extssr 38758 and the comment of df-ssr 38747. (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 6099	. . . . 5 ⊢ ◡ I = I
2	1	eceq2i 8677	. . . 4 ⊢ [𝐴]◡ I = [𝐴] I
3		ecidsn 8693	. . . 4 ⊢ [𝐴] I = {𝐴}
4	2, 3	eqtri 2759	. . 3 ⊢ [𝐴]◡ I = {𝐴}
5	1	eceq2i 8677	. . . 4 ⊢ [𝐵]◡ I = [𝐵] I
6		ecidsn 8693	. . . 4 ⊢ [𝐵] I = {𝐵}
7	5, 6	eqtri 2759	. . 3 ⊢ [𝐵]◡ I = {𝐵}
8	4, 7	eqeq12i 2754	. 2 ⊢ ([𝐴]◡ I = [𝐵]◡ I ↔ {𝐴} = {𝐵})
9		sneqbg 4799	. 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 1541   ∈ wcel 2113  {csn 4580   I cid 5518  ^◡ccnv 5623  [cec 8633

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ec 8637

This theorem is referenced by: (None)