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Theorem bj-seex 34138
Description: Version of seex 5511 with a disjoint variable condition replaced by a nonfreeness hypothesis (for the sake of illustration). (Contributed by BJ, 27-Apr-2019.)
Hypothesis
Ref Expression
bj-seex.nf 𝑥𝐵
Assertion
Ref Expression
bj-seex ((𝑅 Se 𝐴𝐵𝐴) → {𝑥𝐴𝑥𝑅𝐵} ∈ V)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑅
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem bj-seex
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-se 5508 . 2 (𝑅 Se 𝐴 ↔ ∀𝑦𝐴 {𝑥𝐴𝑥𝑅𝑦} ∈ V)
2 bj-seex.nf . . . . . 6 𝑥𝐵
32nfeq2 2992 . . . . 5 𝑥 𝑦 = 𝐵
4 breq2 5061 . . . . 5 (𝑦 = 𝐵 → (𝑥𝑅𝑦𝑥𝑅𝐵))
53, 4rabbid 3473 . . . 4 (𝑦 = 𝐵 → {𝑥𝐴𝑥𝑅𝑦} = {𝑥𝐴𝑥𝑅𝐵})
65eleq1d 2894 . . 3 (𝑦 = 𝐵 → ({𝑥𝐴𝑥𝑅𝑦} ∈ V ↔ {𝑥𝐴𝑥𝑅𝐵} ∈ V))
76rspccva 3619 . 2 ((∀𝑦𝐴 {𝑥𝐴𝑥𝑅𝑦} ∈ V ∧ 𝐵𝐴) → {𝑥𝐴𝑥𝑅𝐵} ∈ V)
81, 7sylanb 581 1 ((𝑅 Se 𝐴𝐵𝐴) → {𝑥𝐴𝑥𝑅𝐵} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wnfc 2958  wral 3135  {crab 3139  Vcvv 3492   class class class wbr 5057   Se wse 5505
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-se 5508
This theorem is referenced by: (None)
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