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Theorem rspc2ev 2664
 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 2656 . . . 4 ((𝐵𝐷𝜓) → ∃𝑦𝐷 𝜒)
32anim2i 324 . . 3 ((𝐴𝐶 ∧ (𝐵𝐷𝜓)) → (𝐴𝐶 ∧ ∃𝑦𝐷 𝜒))
433impb 1100 . 2 ((𝐴𝐶𝐵𝐷𝜓) → (𝐴𝐶 ∧ ∃𝑦𝐷 𝜒))
5 rspc2v.1 . . . 4 (𝑥 = 𝐴 → (𝜑𝜒))
65rexbidv 2327 . . 3 (𝑥 = 𝐴 → (∃𝑦𝐷 𝜑 ↔ ∃𝑦𝐷 𝜒))
76rspcev 2656 . 2 ((𝐴𝐶 ∧ ∃𝑦𝐷 𝜒) → ∃𝑥𝐶𝑦𝐷 𝜑)
84, 7syl 14 1 ((𝐴𝐶𝐵𝐷𝜓) → ∃𝑥𝐶𝑦𝐷 𝜑)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∧ w3a 885   = wceq 1243   ∈ wcel 1393  ∃wrex 2307 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559 This theorem is referenced by:  rspc3ev  2666  opelxp  4374  rspceov  5547  2dom  6285  apreim  7594  addcn2  9831  mulcn2  9833
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