Theorem extid 38637

Description: Property of identity relation, see also extep 38610, extssr 38910 and the comment of df-ssr 38899. (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 6105	. . . . 5 ⊢ ◡ I = I
2	1	eceq2i 8686	. . . 4 ⊢ [𝐴]◡ I = [𝐴] I
3		ecidsn 8702	. . . 4 ⊢ [𝐴] I = {𝐴}
4	2, 3	eqtri 2759	. . 3 ⊢ [𝐴]◡ I = {𝐴}
5	1	eceq2i 8686	. . . 4 ⊢ [𝐵]◡ I = [𝐵] I
6		ecidsn 8702	. . . 4 ⊢ [𝐵] I = {𝐵}
7	5, 6	eqtri 2759	. . 3 ⊢ [𝐵]◡ I = {𝐵}
8	4, 7	eqeq12i 2754	. 2 ⊢ ([𝐴]◡ I = [𝐵]◡ I ↔ {𝐴} = {𝐵})
9		sneqbg 4786	. 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 1542   ∈ wcel 2114  {csn 4567   I cid 5525  ^◡ccnv 5630  [cec 8641

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ec 8645

This theorem is referenced by: (None)