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Theorem rspcedv 3606
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 3604 1 (𝜑 → (𝜒 → ∃𝑥𝐵 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wrex 3071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072
This theorem is referenced by:  rspcebdv  3607  rspcev  3613  rspcedvd  3615  0csh0  14740  gcdcllem1  16437  nn0gsumfz  19847  pmatcollpw3lem  22277  pmatcollpw3fi1lem2  22281  pm2mpfo  22308  f1otrg  28112  cusgrfilem2  28703  wwlksnredwwlkn  29139  wwlksnextprop  29156  clwwlknun  29355  cusconngr  29434  xrofsup  31968  esum2d  33080  rexzrexnn0  41528  onsucelab  41999  ordnexbtwnsuc  42003  ov2ssiunov2  42437  requad2  46278  lcoel0  47063  lcoss  47071  el0ldep  47101  ldepspr  47108  islindeps2  47118  isldepslvec2  47120  affinecomb1  47342
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