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Theorem rspcedv 3577
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 251 . 2 ((𝜑𝑥 = 𝐴) → (𝜒𝜓))
41, 3rspcimedv 3575 1 (𝜑 → (𝜒 → ∃𝑥𝐵 𝜓))
Colors of variables:    wff setvar class
This proof depends on syntax axioms:  wi 4  wb 209  wa 401   = wceq 1570  wcel 2146  wrex 3092
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738
This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-rex 3093
This theorem is used by:  rspcebdv  3578  rspcev  3584  rspcedvd  3586  0csh0  14841  gcdcllem1  16567  nn0gsumfz  20064  pmatcollpw3lem  22955  pmatcollpw3fi1lem2  22959  pm2mpfo  22986  f1otrg  29235  cusgrfilem2  29821  wwlksnredwwlkn  30259  wwlksnextprop  30276  clwwlknun  30478  cusconngr  30557  xrofsup  33127  esum2d  34496  rexzrexnn0  43563  onsucelab  44022  ordnexbtwnsuc  44026  ov2ssiunov2  44458  requad2  48420  lcoel0  49240  lcoss  49248  el0ldep  49278  ldepspr  49285  islindeps2  49295  isldepslvec2  49297  affinecomb1  49514  isisod  49837
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