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Theorem rspcegf 41639
Description: A version of rspcev 3574 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
rspcegf.1 𝑥𝜓
rspcegf.2 𝑥𝐴
rspcegf.3 𝑥𝐵
rspcegf.4 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
rspcegf ((𝐴𝐵𝜓) → ∃𝑥𝐵 𝜑)

Proof of Theorem rspcegf
StepHypRef Expression
1 rspcegf.2 . . . 4 𝑥𝐴
2 rspcegf.3 . . . . . 6 𝑥𝐵
31, 2nfel 2972 . . . . 5 𝑥 𝐴𝐵
4 rspcegf.1 . . . . 5 𝑥𝜓
53, 4nfan 1900 . . . 4 𝑥(𝐴𝐵𝜓)
6 eleq1 2880 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
7 rspcegf.4 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
86, 7anbi12d 633 . . . 4 (𝑥 = 𝐴 → ((𝑥𝐵𝜑) ↔ (𝐴𝐵𝜓)))
91, 5, 8spcegf 3542 . . 3 (𝐴𝐵 → ((𝐴𝐵𝜓) → ∃𝑥(𝑥𝐵𝜑)))
109anabsi5 668 . 2 ((𝐴𝐵𝜓) → ∃𝑥(𝑥𝐵𝜑))
11 df-rex 3115 . 2 (∃𝑥𝐵 𝜑 ↔ ∃𝑥(𝑥𝐵𝜑))
1210, 11sylibr 237 1 ((𝐴𝐵𝜓) → ∃𝑥𝐵 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wex 1781  wnf 1785  wcel 2112  wnfc 2939  wrex 3110
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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773
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 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-rex 3115  df-v 3446
This theorem is referenced by:  rspcef  41693  stoweidlem46  42675
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