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Theorem rspc2v 2748
Description: 2-variable restricted specialization, using implicit substitution. (Contributed by NM, 13-Sep-1999.)
Hypotheses
Ref Expression
rspc2v.1 (𝑥 = 𝐴 → (𝜑𝜒))
rspc2v.2 (𝑦 = 𝐵 → (𝜒𝜓))
Assertion
Ref Expression
rspc2v ((𝐴𝐶𝐵𝐷) → (∀𝑥𝐶𝑦𝐷 𝜑𝜓))
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 rspc2v
StepHypRef Expression
1 nfv 1473 . 2 𝑥𝜒
2 nfv 1473 . 2 𝑦𝜓
3 rspc2v.1 . 2 (𝑥 = 𝐴 → (𝜑𝜒))
4 rspc2v.2 . 2 (𝑦 = 𝐵 → (𝜒𝜓))
51, 2, 3, 4rspc2 2746 1 ((𝐴𝐶𝐵𝐷) → (∀𝑥𝐶𝑦𝐷 𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1296  wcel 1445  wral 2370
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-v 2635
This theorem is referenced by:  rspc2va  2749  rspc3v  2751  disji2  3860  wetriext  4420  f1veqaeq  5586  isorel  5625  fovcl  5788  caovclg  5835  caovcomg  5838  smoel  6103  dcdifsnid  6303  unfiexmid  6708  fiintim  6719  supmoti  6768  supsnti  6780  isotilem  6781  cauappcvgprlem1  7315  caucvgprlemnkj  7322  caucvgprlemnbj  7323  caucvgprprlemval  7344  ltordlem  8057  frecuzrdgrrn  9964  frec2uzrdg  9965  frecuzrdgrcl  9966  frecuzrdgrclt  9971  seq3caopr3  10047  seq3homo  10076  climcn2  10868  inopn  11870  basis1  11913  basis2  11914  xmeteq0  12161  cncfi  12347
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