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Theorem rspcedv 3605
Description: Restricted existential specialization, using implicit substitution. (Contributed by FL, 17-Apr-2007.) (Revised by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
rspcdv.1 (𝜑𝐴𝐵)
rspcdv.2 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
Assertion
Ref Expression
rspcedv (𝜑 → (𝜒 → ∃𝑥𝐵 𝜓))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥   𝜒,𝑥
Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem rspcedv
StepHypRef Expression
1 rspcdv.1 . 2 (𝜑𝐴𝐵)
2 rspcdv.2 . . 3 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
32biimprd 247 . 2 ((𝜑𝑥 = 𝐴) → (𝜒𝜓))
41, 3rspcimedv 3603 1 (𝜑 → (𝜒 → ∃𝑥𝐵 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wrex 3070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071
This theorem is referenced by:  rspcebdv  3606  rspcev  3612  rspcedvd  3614  0csh0  14742  gcdcllem1  16439  nn0gsumfz  19851  pmatcollpw3lem  22284  pmatcollpw3fi1lem2  22288  pm2mpfo  22315  f1otrg  28119  cusgrfilem2  28710  wwlksnredwwlkn  29146  wwlksnextprop  29163  clwwlknun  29362  cusconngr  29441  xrofsup  31975  esum2d  33086  rexzrexnn0  41532  onsucelab  42003  ordnexbtwnsuc  42007  ov2ssiunov2  42441  requad2  46281  lcoel0  47099  lcoss  47107  el0ldep  47137  ldepspr  47144  islindeps2  47154  isldepslvec2  47156  affinecomb1  47378
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