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Theorem extid 36817
Description: Property of identity relation, see also extep 36789, extssr 37017 and the comment of df-ssr 37006. (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 6095 . . . . 5 I = I
21eceq2i 8692 . . . 4 [𝐴] I = [𝐴] I
3 ecidsn 8704 . . . 4 [𝐴] I = {𝐴}
42, 3eqtri 2761 . . 3 [𝐴] I = {𝐴}
51eceq2i 8692 . . . 4 [𝐵] I = [𝐵] I
6 ecidsn 8704 . . . 4 [𝐵] I = {𝐵}
75, 6eqtri 2761 . . 3 [𝐵] I = {𝐵}
84, 7eqeq12i 2751 . 2 ([𝐴] I = [𝐵] I ↔ {𝐴} = {𝐵})
9 sneqbg 4802 . 2 (𝐴𝑉 → ({𝐴} = {𝐵} ↔ 𝐴 = 𝐵))
108, 9bitrid 283 1 (𝐴𝑉 → ([𝐴] I = [𝐵] I ↔ 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1542  wcel 2107  {csn 4587   I cid 5531  ccnv 5633  [cec 8649
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ec 8653
This theorem is referenced by: (None)
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