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Theorem rabexgf 41824
 Description: A version of rabexg 5202 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 3115 . . 3 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
2 simpl 486 . . . . 5 ((𝑥𝐴𝜑) → 𝑥𝐴)
32ss2abi 3996 . . . 4 {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑥𝑥𝐴}
4 rabexgf.1 . . . . 5 𝑥𝐴
54abid2f 2984 . . . 4 {𝑥𝑥𝐴} = 𝐴
63, 5sseqtri 3953 . . 3 {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ 𝐴
71, 6eqsstri 3951 . 2 {𝑥𝐴𝜑} ⊆ 𝐴
8 ssexg 5195 . 2 (({𝑥𝐴𝜑} ⊆ 𝐴𝐴𝑉) → {𝑥𝐴𝜑} ∈ V)
97, 8mpan 689 1 (𝐴𝑉 → {𝑥𝐴𝜑} ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2111  {cab 2776  Ⅎwnfc 2936  {crab 3110  Vcvv 3442   ⊆ wss 3883 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-rab 3115  df-v 3444  df-in 3890  df-ss 3900 This theorem is referenced by:  rabexf  41941  stoweidlem27  42837  stoweidlem35  42845
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