Theorem extid 35583

Description: Property of identity relation, see also extep 35555, extssr 35764 and the comment of df-ssr 35753. (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 6000	. . . . 5 ⊢ ◡ I = I
2	1	eceq2i 8330	. . . 4 ⊢ [𝐴]◡ I = [𝐴] I
3		ecidsn 8342	. . . 4 ⊢ [𝐴] I = {𝐴}
4	2, 3	eqtri 2844	. . 3 ⊢ [𝐴]◡ I = {𝐴}
5	1	eceq2i 8330	. . . 4 ⊢ [𝐵]◡ I = [𝐵] I
6		ecidsn 8342	. . . 4 ⊢ [𝐵] I = {𝐵}
7	5, 6	eqtri 2844	. . 3 ⊢ [𝐵]◡ I = {𝐵}
8	4, 7	eqeq12i 2836	. 2 ⊢ ([𝐴]◡ I = [𝐵]◡ I ↔ {𝐴} = {𝐵})
9		sneqbg 4774	. 2 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} ↔ 𝐴 = 𝐵))
10	8, 9	syl5bb 285	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴]◡ I = [𝐵]◡ I ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1537   ∈ wcel 2114  {csn 4567   I cid 5459  ^◡ccnv 5554  [cec 8287

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-br 5067  df-opab 5129  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-ec 8291

This theorem is referenced by: (None)