Theorem extid 37167

Description: Property of identity relation, see also extep 37139, extssr 37367 and the comment of df-ssr 37356. (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 6138	. . . . 5 ⊢ ◡ I = I
2	1	eceq2i 8740	. . . 4 ⊢ [𝐴]◡ I = [𝐴] I
3		ecidsn 8752	. . . 4 ⊢ [𝐴] I = {𝐴}
4	2, 3	eqtri 2760	. . 3 ⊢ [𝐴]◡ I = {𝐴}
5	1	eceq2i 8740	. . . 4 ⊢ [𝐵]◡ I = [𝐵] I
6		ecidsn 8752	. . . 4 ⊢ [𝐵] I = {𝐵}
7	5, 6	eqtri 2760	. . 3 ⊢ [𝐵]◡ I = {𝐵}
8	4, 7	eqeq12i 2750	. 2 ⊢ ([𝐴]◡ I = [𝐵]◡ I ↔ {𝐴} = {𝐵})
9		sneqbg 4843	. 2 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} ↔ 𝐴 = 𝐵))
10	8, 9	bitrid 282	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴]◡ I = [𝐵]◡ I ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1541   ∈ wcel 2106  {csn 4627   I cid 5572  ^◡ccnv 5674  [cec 8697

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ec 8701

This theorem is referenced by: (None)