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Theorem rspcef 45011
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 44960 1 ((𝐴𝐵𝜓) → ∃𝑥𝐵 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wnf 1779  wcel 2105  wnfc 2887  wrex 3067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1539  df-ex 1776  df-nf 1780  df-cleq 2726  df-clel 2813  df-nfc 2889  df-rex 3068
This theorem is referenced by:  iinssdf  45078  rspced  45109  opnvonmbllem1  46587  smfresal  46743  smfmullem2  46747
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