Theorem extid 38293

Description: Property of identity relation, see also extep 38266, extssr 38495 and the comment of df-ssr 38484. (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 6116	. . . . 5 ⊢ ◡ I = I
2	1	eceq2i 8715	. . . 4 ⊢ [𝐴]◡ I = [𝐴] I
3		ecidsn 8731	. . . 4 ⊢ [𝐴] I = {𝐴}
4	2, 3	eqtri 2753	. . 3 ⊢ [𝐴]◡ I = {𝐴}
5	1	eceq2i 8715	. . . 4 ⊢ [𝐵]◡ I = [𝐵] I
6		ecidsn 8731	. . . 4 ⊢ [𝐵] I = {𝐵}
7	5, 6	eqtri 2753	. . 3 ⊢ [𝐵]◡ I = {𝐵}
8	4, 7	eqeq12i 2748	. 2 ⊢ ([𝐴]◡ I = [𝐵]◡ I ↔ {𝐴} = {𝐵})
9		sneqbg 4809	. 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 1540   ∈ wcel 2109  {csn 4591   I cid 5534  ^◡ccnv 5639  [cec 8671

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pr 5389

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-br 5110  df-opab 5172  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-ec 8675

This theorem is referenced by: (None)