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Theorem extep 35700
Description: Property of epsilon relation, see also extid 35728, extssr 35909 and the comment of df-ssr 35898. (Contributed by Peter Mazsa, 10-Jul-2019.)
Assertion
Ref Expression
extep ((𝐴𝑉𝐵𝑊) → ([𝐴] E = [𝐵] E ↔ 𝐴 = 𝐵))

Proof of Theorem extep
StepHypRef Expression
1 eccnvep 35699 . 2 (𝐴𝑉 → [𝐴] E = 𝐴)
2 eccnvep 35699 . 2 (𝐵𝑊 → [𝐵] E = 𝐵)
31, 2eqeqan12d 2815 1 ((𝐴𝑉𝐵𝑊) → ([𝐴] E = [𝐵] E ↔ 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2111   E cep 5429  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-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  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-eprel 5430  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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