Theorem extid 38266

Description: Property of identity relation, see also extep 38239, extssr 38465 and the comment of df-ssr 38454. (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 6173	. . . . 5 ⊢ ◡ I = I
2	1	eceq2i 8805	. . . 4 ⊢ [𝐴]◡ I = [𝐴] I
3		ecidsn 8818	. . . 4 ⊢ [𝐴] I = {𝐴}
4	2, 3	eqtri 2768	. . 3 ⊢ [𝐴]◡ I = {𝐴}
5	1	eceq2i 8805	. . . 4 ⊢ [𝐵]◡ I = [𝐵] I
6		ecidsn 8818	. . . 4 ⊢ [𝐵] I = {𝐵}
7	5, 6	eqtri 2768	. . 3 ⊢ [𝐵]◡ I = {𝐵}
8	4, 7	eqeq12i 2758	. 2 ⊢ ([𝐴]◡ I = [𝐵]◡ I ↔ {𝐴} = {𝐵})
9		sneqbg 4868	. 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 1537   ∈ wcel 2108  {csn 4648   I cid 5592  ^◡ccnv 5699  [cec 8761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ec 8765

This theorem is referenced by: (None)