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Theorem bj-seex 35740
Description: Version of seex 5637 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 5631 . 2 (𝑅 Se 𝐴 ↔ ∀𝑦𝐴 {𝑥𝐴𝑥𝑅𝑦} ∈ V)
2 bj-seex.nf . . . . . 6 𝑥𝐵
32nfeq2 2921 . . . . 5 𝑥 𝑦 = 𝐵
4 breq2 5151 . . . . 5 (𝑦 = 𝐵 → (𝑥𝑅𝑦𝑥𝑅𝐵))
53, 4rabbid 3460 . . . 4 (𝑦 = 𝐵 → {𝑥𝐴𝑥𝑅𝑦} = {𝑥𝐴𝑥𝑅𝐵})
65eleq1d 2819 . . 3 (𝑦 = 𝐵 → ({𝑥𝐴𝑥𝑅𝑦} ∈ V ↔ {𝑥𝐴𝑥𝑅𝐵} ∈ V))
76rspccva 3611 . 2 ((∀𝑦𝐴 {𝑥𝐴𝑥𝑅𝑦} ∈ V ∧ 𝐵𝐴) → {𝑥𝐴𝑥𝑅𝐵} ∈ V)
81, 7sylanb 582 1 ((𝑅 Se 𝐴𝐵𝐴) → {𝑥𝐴𝑥𝑅𝐵} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wnfc 2884  wral 3062  {crab 3433  Vcvv 3475   class class class wbr 5147   Se wse 5628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-se 5631
This theorem is referenced by: (None)
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