Theorem extid 38283

Description: Property of identity relation, see also extep 38256, extssr 38485 and the comment of df-ssr 38474. (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 6094	. . . . 5 ⊢ ◡ I = I
2	1	eceq2i 8674	. . . 4 ⊢ [𝐴]◡ I = [𝐴] I
3		ecidsn 8690	. . . 4 ⊢ [𝐴] I = {𝐴}
4	2, 3	eqtri 2752	. . 3 ⊢ [𝐴]◡ I = {𝐴}
5	1	eceq2i 8674	. . . 4 ⊢ [𝐵]◡ I = [𝐵] I
6		ecidsn 8690	. . . 4 ⊢ [𝐵] I = {𝐵}
7	5, 6	eqtri 2752	. . 3 ⊢ [𝐵]◡ I = {𝐵}
8	4, 7	eqeq12i 2747	. 2 ⊢ ([𝐴]◡ I = [𝐵]◡ I ↔ {𝐴} = {𝐵})
9		sneqbg 4797	. 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 4579   I cid 5517  ^◡ccnv 5622  [cec 8630

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 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

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 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096  df-opab 5158  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-ec 8634

This theorem is referenced by: (None)