Theorem extid 38343

Description: Property of identity relation, see also extep 38316, extssr 38545 and the comment of df-ssr 38534. (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 6088	. . . . 5 ⊢ ◡ I = I
2	1	eceq2i 8664	. . . 4 ⊢ [𝐴]◡ I = [𝐴] I
3		ecidsn 8680	. . . 4 ⊢ [𝐴] I = {𝐴}
4	2, 3	eqtri 2754	. . 3 ⊢ [𝐴]◡ I = {𝐴}
5	1	eceq2i 8664	. . . 4 ⊢ [𝐵]◡ I = [𝐵] I
6		ecidsn 8680	. . . 4 ⊢ [𝐵] I = {𝐵}
7	5, 6	eqtri 2754	. . 3 ⊢ [𝐵]◡ I = {𝐵}
8	4, 7	eqeq12i 2749	. 2 ⊢ ([𝐴]◡ I = [𝐵]◡ I ↔ {𝐴} = {𝐵})
9		sneqbg 4795	. 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 1541   ∈ wcel 2111  {csn 4576   I cid 5510  ^◡ccnv 5615  [cec 8620

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-br 5092  df-opab 5154  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-ec 8624

This theorem is referenced by: (None)