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Theorem rspcef 39732
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 39673 1 ((𝐴𝐵𝜓) → ∃𝑥𝐵 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1624  wnf 1849  wcel 2131  wnfc 2881  wrex 3043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-rex 3048  df-v 3334
This theorem is referenced by:  iinssdf  39819  opnvonmbllem1  41344  smfresal  41493  smfmullem2  41497
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