Theorem extid 36446

Description: Property of identity relation, see also extep 36418, extssr 36627 and the comment of df-ssr 36616. (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 6045	. . . . 5 ⊢ ◡ I = I
2	1	eceq2i 8539	. . . 4 ⊢ [𝐴]◡ I = [𝐴] I
3		ecidsn 8551	. . . 4 ⊢ [𝐴] I = {𝐴}
4	2, 3	eqtri 2766	. . 3 ⊢ [𝐴]◡ I = {𝐴}
5	1	eceq2i 8539	. . . 4 ⊢ [𝐵]◡ I = [𝐵] I
6		ecidsn 8551	. . . 4 ⊢ [𝐵] I = {𝐵}
7	5, 6	eqtri 2766	. . 3 ⊢ [𝐵]◡ I = {𝐵}
8	4, 7	eqeq12i 2756	. 2 ⊢ ([𝐴]◡ I = [𝐵]◡ I ↔ {𝐴} = {𝐵})
9		sneqbg 4774	. 2 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} ↔ 𝐴 = 𝐵))
10	8, 9	syl5bb 283	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴]◡ I = [𝐵]◡ I ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539   ∈ wcel 2106  {csn 4561   I cid 5488  ^◡ccnv 5588  [cec 8496

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ec 8500

This theorem is referenced by: (None)