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Theorem rspcef 41211
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.1 . 2 𝑥𝜓
2 rspcef.2 . 2 𝑥𝐴
3 rspcef.3 . 2 𝑥𝐵
4 rspcef.4 . 2 (𝑥 = 𝐴 → (𝜑𝜓))
51, 2, 3, 4rspcegf 41157 1 ((𝐴𝐵𝜓) → ∃𝑥𝐵 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wnf 1775  wcel 2105  wnfc 2958  wrex 3136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rex 3141  df-v 3494
This theorem is referenced by:  iinssdf  41284  opnvonmbllem1  42791  smfresal  42940  smfmullem2  42944
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