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Theorem rspc2va 3593
Description: 2-variable restricted specialization, using implicit substitution. (Contributed by NM, 18-Jun-2014.)
Hypotheses
Ref Expression
rspc2v.1 (𝑥 = 𝐴 → (𝜑𝜒))
rspc2v.2 (𝑦 = 𝐵 → (𝜒𝜓))
Assertion
Ref Expression
rspc2va (((𝐴𝐶𝐵𝐷) ∧ ∀𝑥𝐶𝑦𝐷 𝜑) → 𝜓)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵   𝑥,𝐶   𝑥,𝐷,𝑦   𝜒,𝑥   𝜓,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥)   𝜒(𝑦)   𝐵(𝑥)   𝐶(𝑦)

Proof of Theorem rspc2va
StepHypRef Expression
1 rspc2v.1 . . 3 (𝑥 = 𝐴 → (𝜑𝜒))
2 rspc2v.2 . . 3 (𝑦 = 𝐵 → (𝜒𝜓))
31, 2rspc2v 3592 . 2 ((𝐴𝐶𝐵𝐷) → (∀𝑥𝐶𝑦𝐷 𝜑𝜓))
43imp 408 1 (((𝐴𝐶𝐵𝐷) ∧ ∀𝑥𝐶𝑦𝐷 𝜑) → 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3061
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 3062
This theorem is referenced by:  swopo  5560  soisores  7276  soisoi  7277  isocnv  7279  isotr  7285  ovrspc2v  7387  caofrss  7657  caonncan  7662  frpoins3xpg  8076  coflton  8621  wunpr  10653  injresinj  13702  seqcaopr2  13953  rlimcn3  15481  o1of2  15504  isprm6  16598  ssc2  17713  pospropd  18224  tleile  18318  mhmpropd  18616  grpidssd  18831  grpinvssd  18832  dfgrp3lem  18853  isnsg3  18970  cyccom  19004  symgextf1  19211  efgredlemd  19534  efgredlem  19537  rglcom4d  19950  issrngd  20363  domneq0  20790  lindfind  21245  lindsind  21246  mplsubglem  21428  mdetunilem1  21984  mdetunilem3  21986  mdetunilem4  21987  mdetunilem9  21992  decpmatmulsumfsupp  22145  pm2mpf1  22171  pm2mpmhmlem1  22190  t0sep  22698  tsmsxplem2  23528  comet  23892  nrmmetd  23953  tngngp  24041  reconnlem2  24213  iscmet3lem1  24678  iscmet3lem2  24679  dchrisumlem1  26860  pntpbnd1  26957  ssltsepc  27161  tgjustc1  27466  tgjustc2  27467  iscgrglt  27505  motcgr  27527  perpneq  27705  foot  27713  f1otrg  27862  axcontlem10  27971  frgr2wwlk1  29322  orngmul  32152  lindsunlem  32383  mndpluscn  32571  unelros  32834  difelros  32835  inelsros  32841  diffiunisros  32842  cvxsconn  33901  elmrsubrn  34178  ghomco  36400  sticksstones10  40613  sticksstones12a  40615  fsuppind  40812  mzpcl34  41101  ntrk0kbimka  42403  isotone1  42412  isotone2  42413  nnfoctbdjlem  44786  isomuspgrlem2d  46113  mgmhmpropd  46169  rnghmmul  46288  2arymaptf1  46829
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