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Theorem rabexgf 39678
Description: A version of rabexg 4959 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 3055 . . 3 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
2 simpl 474 . . . . 5 ((𝑥𝐴𝜑) → 𝑥𝐴)
32ss2abi 3811 . . . 4 {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑥𝑥𝐴}
4 rabexgf.1 . . . . 5 𝑥𝐴
54abid2f 2925 . . . 4 {𝑥𝑥𝐴} = 𝐴
63, 5sseqtri 3774 . . 3 {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ 𝐴
71, 6eqsstri 3772 . 2 {𝑥𝐴𝜑} ⊆ 𝐴
8 ssexg 4952 . 2 (({𝑥𝐴𝜑} ⊆ 𝐴𝐴𝑉) → {𝑥𝐴𝜑} ∈ V)
97, 8mpan 708 1 (𝐴𝑉 → {𝑥𝐴𝜑} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 2135  {cab 2742  wnfc 2885  {crab 3050  Vcvv 3336  wss 3711
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-sep 4929
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-rab 3055  df-v 3338  df-in 3718  df-ss 3725
This theorem is referenced by:  rabexf  39814  stoweidlem27  40743  stoweidlem35  40751
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