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Theorem rspc2v 2802
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 1508 . 2 𝑥𝜒
2 nfv 1508 . 2 𝑦𝜓
3 rspc2v.1 . 2 (𝑥 = 𝐴 → (𝜑𝜒))
4 rspc2v.2 . 2 (𝑦 = 𝐵 → (𝜒𝜓))
51, 2, 3, 4rspc2 2800 1 ((𝐴𝐶𝐵𝐷) → (∀𝑥𝐶𝑦𝐷 𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1331  wcel 1480  wral 2416
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688
This theorem is referenced by:  rspc2va  2803  rspc3v  2805  disji2  3922  wetriext  4491  f1veqaeq  5670  isorel  5709  fovcl  5876  caovclg  5923  caovcomg  5926  smoel  6197  dcdifsnid  6400  unfiexmid  6806  fiintim  6817  supmoti  6880  supsnti  6892  isotilem  6893  cauappcvgprlem1  7474  caucvgprlemnkj  7481  caucvgprlemnbj  7482  caucvgprprlemval  7503  ltordlem  8251  frecuzrdgrrn  10188  frec2uzrdg  10189  frecuzrdgrcl  10190  frecuzrdgrclt  10195  seq3caopr3  10261  seq3homo  10290  climcn2  11085  ennnfonelemim  11944  inopn  12180  basis1  12224  basis2  12225  xmeteq0  12538  cncfi  12744  limccnp2lem  12824
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