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Theorem rabexgf 38663
Description: A version of rabexg 4772 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 2916 . . 3 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
2 simpl 473 . . . . 5 ((𝑥𝐴𝜑) → 𝑥𝐴)
32ss2abi 3653 . . . 4 {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑥𝑥𝐴}
4 rabexgf.1 . . . . 5 𝑥𝐴
54abid2f 2787 . . . 4 {𝑥𝑥𝐴} = 𝐴
63, 5sseqtri 3616 . . 3 {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ 𝐴
71, 6eqsstri 3614 . 2 {𝑥𝐴𝜑} ⊆ 𝐴
8 ssexg 4764 . 2 (({𝑥𝐴𝜑} ⊆ 𝐴𝐴𝑉) → {𝑥𝐴𝜑} ∈ V)
97, 8mpan 705 1 (𝐴𝑉 → {𝑥𝐴𝜑} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1987  {cab 2607  wnfc 2748  {crab 2911  Vcvv 3186  wss 3555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2916  df-v 3188  df-in 3562  df-ss 3569
This theorem is referenced by:  rabexf  38805  stoweidlem27  39548  stoweidlem35  39556
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