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Theorem rabexgf 38206
 Description: A version of rabexg 4739 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypothesis
Ref Expression
rabexgf.1 𝑥𝐴
Assertion
Ref Expression
rabexgf (𝐴𝑉 → {𝑥𝐴𝜑} ∈ V)

Proof of Theorem rabexgf
StepHypRef Expression
1 df-rab 2905 . . 3 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
2 simpl 472 . . . . 5 ((𝑥𝐴𝜑) → 𝑥𝐴)
32ss2abi 3637 . . . 4 {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑥𝑥𝐴}
4 rabexgf.1 . . . . 5 𝑥𝐴
54abid2f 2777 . . . 4 {𝑥𝑥𝐴} = 𝐴
63, 5sseqtri 3600 . . 3 {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ 𝐴
71, 6eqsstri 3598 . 2 {𝑥𝐴𝜑} ⊆ 𝐴
8 ssexg 4732 . 2 (({𝑥𝐴𝜑} ⊆ 𝐴𝐴𝑉) → {𝑥𝐴𝜑} ∈ V)
97, 8mpan 702 1 (𝐴𝑉 → {𝑥𝐴𝜑} ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 1977  {cab 2596  Ⅎwnfc 2738  {crab 2900  Vcvv 3173   ⊆ wss 3540 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-in 3547  df-ss 3554 This theorem is referenced by:  stoweidlem27  38920  stoweidlem35  38928
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