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Theorem rspcef 38267
 Description: Restricted existential specialization, using implicit substitution. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
Hypotheses
Ref Expression
rspcef.1 𝑥𝜓
rspcef.2 𝑥𝐴
rspcef.3 𝑥𝐵
rspcef.4 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
rspcef ((𝐴𝐵𝜓) → ∃𝑥𝐵 𝜑)

Proof of Theorem rspcef
StepHypRef Expression
1 rspcef.2 . . . 4 𝑥𝐴
2 rspcef.3 . . . . . 6 𝑥𝐵
31, 2nfel 2763 . . . . 5 𝑥 𝐴𝐵
4 rspcef.1 . . . . 5 𝑥𝜓
53, 4nfan 1816 . . . 4 𝑥(𝐴𝐵𝜓)
6 eleq1 2676 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
7 rspcef.4 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
86, 7anbi12d 743 . . . 4 (𝑥 = 𝐴 → ((𝑥𝐵𝜑) ↔ (𝐴𝐵𝜓)))
91, 5, 8spcegf 3262 . . 3 (𝐴𝐵 → ((𝐴𝐵𝜓) → ∃𝑥(𝑥𝐵𝜑)))
109anabsi5 854 . 2 ((𝐴𝐵𝜓) → ∃𝑥(𝑥𝐵𝜑))
11 df-rex 2902 . 2 (∃𝑥𝐵 𝜑 ↔ ∃𝑥(𝑥𝐵𝜑))
1210, 11sylibr 223 1 ((𝐴𝐵𝜓) → ∃𝑥𝐵 𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695  Ⅎwnf 1699   ∈ wcel 1977  Ⅎwnfc 2738  ∃wrex 2897 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-v 3175 This theorem is referenced by:  opnvonmbllem1  39522  smfresal  39673  smfmullem2  39677
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