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Theorem ideqg 4430
Description: For sets, the identity relation is the same as equality. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
ideqg (B 𝑉 → (A I BA = B))

Proof of Theorem ideqg
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 reli 4408 . . . . 5 Rel I
21brrelexi 4327 . . . 4 (A I BA V)
32adantl 262 . . 3 ((B 𝑉 A I B) → A V)
4 simpl 102 . . 3 ((B 𝑉 A I B) → B 𝑉)
53, 4jca 290 . 2 ((B 𝑉 A I B) → (A V B 𝑉))
6 eleq1 2097 . . . . 5 (A = B → (A 𝑉B 𝑉))
76biimparc 283 . . . 4 ((B 𝑉 A = B) → A 𝑉)
8 elex 2560 . . . 4 (A 𝑉A V)
97, 8syl 14 . . 3 ((B 𝑉 A = B) → A V)
10 simpl 102 . . 3 ((B 𝑉 A = B) → B 𝑉)
119, 10jca 290 . 2 ((B 𝑉 A = B) → (A V B 𝑉))
12 eqeq1 2043 . . 3 (x = A → (x = yA = y))
13 eqeq2 2046 . . 3 (y = B → (A = yA = B))
14 df-id 4021 . . 3 I = {⟨x, y⟩ ∣ x = y}
1512, 13, 14brabg 3997 . 2 ((A V B 𝑉) → (A I BA = B))
165, 11, 15pm5.21nd 824 1 (B 𝑉 → (A I BA = B))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390  Vcvv 2551   class class class wbr 3755   I cid 4016
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295
This theorem is referenced by:  ideq  4431  ididg  4432  poleloe  4667
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