Theorem extid 36373

Description: Property of identity relation, see also extep 36345, extssr 36554 and the comment of df-ssr 36543. (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 6034	. . . . 5 ⊢ ◡ I = I
2	1	eceq2i 8497	. . . 4 ⊢ [𝐴]◡ I = [𝐴] I
3		ecidsn 8509	. . . 4 ⊢ [𝐴] I = {𝐴}
4	2, 3	eqtri 2766	. . 3 ⊢ [𝐴]◡ I = {𝐴}
5	1	eceq2i 8497	. . . 4 ⊢ [𝐵]◡ I = [𝐵] I
6		ecidsn 8509	. . . 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 4771	. 2 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} ↔ 𝐴 = 𝐵))
10	8, 9	syl5bb 282	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴]◡ I = [𝐵]◡ I ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539   ∈ wcel 2108  {csn 4558   I cid 5479  ^◡ccnv 5579  [cec 8454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ec 8458

This theorem is referenced by: (None)