Theorem extid 38683

Description: Property of identity relation, see also extep 38656, extssr 38956 and the comment of df-ssr 38945. (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 6092	. . . . 5 ⊢ ◡ I = I
2	1	eceq2i 8676	. . . 4 ⊢ [𝐴]◡ I = [𝐴] I
3		ecidsn 8692	. . . 4 ⊢ [𝐴] I = {𝐴}
4	2, 3	eqtri 2762	. . 3 ⊢ [𝐴]◡ I = {𝐴}
5	1	eceq2i 8676	. . . 4 ⊢ [𝐵]◡ I = [𝐵] I
6		ecidsn 8692	. . . 4 ⊢ [𝐵] I = {𝐵}
7	5, 6	eqtri 2762	. . 3 ⊢ [𝐵]◡ I = {𝐵}
8	4, 7	eqeq12i 2757	. 2 ⊢ ([𝐴]◡ I = [𝐵]◡ I ↔ {𝐴} = {𝐵})
9		sneqbg 4774	. 2 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} ↔ 𝐴 = 𝐵))
10	8, 9	bitrid 284	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴]◡ I = [𝐵]◡ I ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 207   = wceq 1547   ∈ wcel 2119  {csn 4555   I cid 5512  ^◡ccnv 5617  [cec 8631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073  df-opab 5135  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-ec 8635

This theorem is referenced by: (None)