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Theorem bj-unexg 10414
Description: unexg 4205 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-unexg ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)

Proof of Theorem bj-unexg
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uneq1 3117 . . 3 (𝑥 = 𝐴 → (𝑥𝑦) = (𝐴𝑦))
2 eleq1 2116 . . 3 ((𝑥𝑦) = (𝐴𝑦) → ((𝑥𝑦) ∈ V ↔ (𝐴𝑦) ∈ V))
31, 2syl 14 . 2 (𝑥 = 𝐴 → ((𝑥𝑦) ∈ V ↔ (𝐴𝑦) ∈ V))
4 uneq2 3118 . . 3 (𝑦 = 𝐵 → (𝐴𝑦) = (𝐴𝐵))
5 eleq1 2116 . . 3 ((𝐴𝑦) = (𝐴𝐵) → ((𝐴𝑦) ∈ V ↔ (𝐴𝐵) ∈ V))
64, 5syl 14 . 2 (𝑦 = 𝐵 → ((𝐴𝑦) ∈ V ↔ (𝐴𝐵) ∈ V))
7 vex 2577 . . 3 𝑥 ∈ V
8 vex 2577 . . 3 𝑦 ∈ V
97, 8bj-unex 10412 . 2 (𝑥𝑦) ∈ V
103, 6, 9vtocl2g 2634 1 ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wcel 1409  Vcvv 2574  cun 2942
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-pr 3971  ax-un 4197  ax-bd0 10306  ax-bdor 10309  ax-bdex 10312  ax-bdeq 10313  ax-bdel 10314  ax-bdsb 10315  ax-bdsep 10377
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-v 2576  df-un 2949  df-sn 3408  df-pr 3409  df-uni 3608  df-bdc 10334
This theorem is referenced by:  bj-sucexg  10415
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