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Theorem rspc2ev 2725
Description: 2-variable restricted existential specialization, using implicit substitution. (Contributed by NM, 16-Oct-1999.)
Hypotheses
Ref Expression
rspc2v.1 (𝑥 = 𝐴 → (𝜑𝜒))
rspc2v.2 (𝑦 = 𝐵 → (𝜒𝜓))
Assertion
Ref Expression
rspc2ev ((𝐴𝐶𝐵𝐷𝜓) → ∃𝑥𝐶𝑦𝐷 𝜑)
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 rspc2ev
StepHypRef Expression
1 rspc2v.2 . . . . 5 (𝑦 = 𝐵 → (𝜒𝜓))
21rspcev 2712 . . . 4 ((𝐵𝐷𝜓) → ∃𝑦𝐷 𝜒)
32anim2i 334 . . 3 ((𝐴𝐶 ∧ (𝐵𝐷𝜓)) → (𝐴𝐶 ∧ ∃𝑦𝐷 𝜒))
433impb 1135 . 2 ((𝐴𝐶𝐵𝐷𝜓) → (𝐴𝐶 ∧ ∃𝑦𝐷 𝜒))
5 rspc2v.1 . . . 4 (𝑥 = 𝐴 → (𝜑𝜒))
65rexbidv 2375 . . 3 (𝑥 = 𝐴 → (∃𝑦𝐷 𝜑 ↔ ∃𝑦𝐷 𝜒))
76rspcev 2712 . 2 ((𝐴𝐶 ∧ ∃𝑦𝐷 𝜒) → ∃𝑥𝐶𝑦𝐷 𝜑)
84, 7syl 14 1 ((𝐴𝐶𝐵𝐷𝜓) → ∃𝑥𝐶𝑦𝐷 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  w3a 920   = wceq 1285  wcel 1434  wrex 2354
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rex 2359  df-v 2614
This theorem is referenced by:  rspc3ev  2727  opelxp  4430  rspceov  5626  2dom  6452  apreim  7980  addcn2  10523  mulcn2  10525  divalglemnn  10698  bezoutlema  10768  bezoutlemb  10769
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