Theorem extid 37782

Description: Property of identity relation, see also extep 37755, extssr 37981 and the comment of df-ssr 37970. (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 6146	. . . . 5 ⊢ ◡ I = I
2	1	eceq2i 8765	. . . 4 ⊢ [𝐴]◡ I = [𝐴] I
3		ecidsn 8778	. . . 4 ⊢ [𝐴] I = {𝐴}
4	2, 3	eqtri 2756	. . 3 ⊢ [𝐴]◡ I = {𝐴}
5	1	eceq2i 8765	. . . 4 ⊢ [𝐵]◡ I = [𝐵] I
6		ecidsn 8778	. . . 4 ⊢ [𝐵] I = {𝐵}
7	5, 6	eqtri 2756	. . 3 ⊢ [𝐵]◡ I = {𝐵}
8	4, 7	eqeq12i 2746	. 2 ⊢ ([𝐴]◡ I = [𝐵]◡ I ↔ {𝐴} = {𝐵})
9		sneqbg 4845	. 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 205   = wceq 1534   ∈ wcel 2099  {csn 4629   I cid 5575  ^◡ccnv 5677  [cec 8722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ec 8726

This theorem is referenced by: (None)