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Theorem bj-prexg 7281
 Description: Proof of prexg 3921 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-prexg ((A 𝑉 B 𝑊) → {A, B} V)

Proof of Theorem bj-prexg
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 preq2 3422 . . . . . 6 (y = B → {x, y} = {x, B})
21eleq1d 2088 . . . . 5 (y = B → ({x, y} V ↔ {x, B} V))
3 bj-zfpair2 7280 . . . . 5 {x, y} V
42, 3vtoclg 2590 . . . 4 (B 𝑊 → {x, B} V)
5 preq1 3421 . . . . 5 (x = A → {x, B} = {A, B})
65eleq1d 2088 . . . 4 (x = A → ({x, B} V ↔ {A, B} V))
74, 6syl5ib 143 . . 3 (x = A → (B 𝑊 → {A, B} V))
87vtocleg 2601 . 2 (A 𝑉 → (B 𝑊 → {A, B} V))
98imp 115 1 ((A 𝑉 B 𝑊) → {A, B} V)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1228   ∈ wcel 1374  Vcvv 2535  {cpr 3351 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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-pr 3918  ax-bdor 7190  ax-bdeq 7194  ax-bdsep 7258 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-un 2899  df-sn 3356  df-pr 3357 This theorem is referenced by:  bj-snexg  7282  bj-unex  7289
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