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Theorem rspc2va 2757
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 2756 . 2 ((𝐴𝐶𝐵𝐷) → (∀𝑥𝐶𝑦𝐷 𝜑𝜓))
43imp 123 1 (((𝐴𝐶𝐵𝐷) ∧ ∀𝑥𝐶𝑦𝐷 𝜑) → 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1299  wcel 1448  wral 2375
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 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-v 2643
This theorem is referenced by:  swopo  4166  ordtri2orexmid  4376  onsucelsucexmid  4383  ordsucunielexmid  4384  ordtri2or2exmid  4424  isocnv  5644  isotr  5649  off  5926  caofrss  5937  tridc  6722  fidcenumlemrks  6769  seq3caopr2  10096  seq3distr  10127  isprm6  11618  comet  12427  mulcncf  12503  trilpo  12820
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