MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rspcedv Structured version   Visualization version   GIF version

Theorem rspcedv 3614
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 248 . 2 ((𝜑𝑥 = 𝐴) → (𝜒𝜓))
41, 3rspcimedv 3612 1 (𝜑 → (𝜒 → ∃𝑥𝐵 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  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-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1539  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068
This theorem is referenced by:  rspcebdv  3615  rspcev  3621  rspcedvd  3623  0csh0  14827  gcdcllem1  16532  nn0gsumfz  20016  pmatcollpw3lem  22804  pmatcollpw3fi1lem2  22808  pm2mpfo  22835  f1otrg  28893  cusgrfilem2  29488  wwlksnredwwlkn  29924  wwlksnextprop  29941  clwwlknun  30140  cusconngr  30219  xrofsup  32777  esum2d  34073  rexzrexnn0  42791  onsucelab  43252  ordnexbtwnsuc  43256  ov2ssiunov2  43689  requad2  47547  lcoel0  48273  lcoss  48281  el0ldep  48311  ldepspr  48318  islindeps2  48328  isldepslvec2  48330  affinecomb1  48551  isisod  48806
  Copyright terms: Public domain W3C validator