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Theorem rspcegf 44961
Description: A version of rspcev 3622 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 2918 . . . . 5 𝑥 𝐴𝐵
4 rspcegf.1 . . . . 5 𝑥𝜓
53, 4nfan 1897 . . . 4 𝑥(𝐴𝐵𝜓)
6 eleq1 2827 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
7 rspcegf.4 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
86, 7anbi12d 632 . . . 4 (𝑥 = 𝐴 → ((𝑥𝐵𝜑) ↔ (𝐴𝐵𝜓)))
91, 5, 8spcegf 3592 . . 3 (𝐴𝐵 → ((𝐴𝐵𝜓) → ∃𝑥(𝑥𝐵𝜑)))
109anabsi5 669 . 2 ((𝐴𝐵𝜓) → ∃𝑥(𝑥𝐵𝜑))
11 df-rex 3069 . 2 (∃𝑥𝐵 𝜑 ↔ ∃𝑥(𝑥𝐵𝜑))
1210, 11sylibr 234 1 ((𝐴𝐵𝜓) → ∃𝑥𝐵 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wex 1776  wnf 1780  wcel 2106  wnfc 2888  wrex 3068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-cleq 2727  df-clel 2814  df-nfc 2890  df-rex 3069
This theorem is referenced by:  rspcef  45012  stoweidlem46  46002
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