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Theorem rspc2v 2685
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 1437 . 2 𝑥𝜒
2 nfv 1437 . 2 𝑦𝜓
3 rspc2v.1 . 2 (𝑥 = 𝐴 → (𝜑𝜒))
4 rspc2v.2 . 2 (𝑦 = 𝐵 → (𝜒𝜓))
51, 2, 3, 4rspc2 2683 1 ((𝐴𝐶𝐵𝐷) → (∀𝑥𝐶𝑦𝐷 𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wcel 1409  wral 2323
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576
This theorem is referenced by:  rspc2va  2686  rspc3v  2688  wetriext  4329  f1veqaeq  5436  isorel  5476  fovcl  5634  caovclg  5681  caovcomg  5684  smoel  5946  supmoti  6399  supsnti  6409  isotilem  6410  cauappcvgprlem1  6815  caucvgprlemnkj  6822  caucvgprlemnbj  6823  caucvgprprlemval  6844  frecuzrdgrrn  9358  iseqcaopr3  9404  iseqhomo  9412  climcn2  10061
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