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Theorem extid 35728
Description: Property of identity relation, see also extep 35700, extssr 35909 and the comment of df-ssr 35898. (Contributed by Peter Mazsa, 5-Jul-2019.)
Assertion
Ref Expression
extid (𝐴𝑉 → ([𝐴] I = [𝐵] I ↔ 𝐴 = 𝐵))

Proof of Theorem extid
StepHypRef Expression
1 cnvi 5967 . . . . 5 I = I
21eceq2i 8313 . . . 4 [𝐴] I = [𝐴] I
3 ecidsn 8325 . . . 4 [𝐴] I = {𝐴}
42, 3eqtri 2821 . . 3 [𝐴] I = {𝐴}
51eceq2i 8313 . . . 4 [𝐵] I = [𝐵] I
6 ecidsn 8325 . . . 4 [𝐵] I = {𝐵}
75, 6eqtri 2821 . . 3 [𝐵] I = {𝐵}
84, 7eqeq12i 2813 . 2 ([𝐴] I = [𝐵] I ↔ {𝐴} = {𝐵})
9 sneqbg 4734 . 2 (𝐴𝑉 → ({𝐴} = {𝐵} ↔ 𝐴 = 𝐵))
108, 9syl5bb 286 1 (𝐴𝑉 → ([𝐴] I = [𝐵] I ↔ 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1538  wcel 2111  {csn 4525   I cid 5424  ccnv 5518  [cec 8270
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-ec 8274
This theorem is referenced by: (None)
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