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Theorem rspce 2758
Description: Restricted existential specialization, using implicit substitution. (Contributed by NM, 26-May-1998.) (Revised by Mario Carneiro, 11-Oct-2016.)
Hypotheses
Ref Expression
rspc.1 𝑥𝜓
rspc.2 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
rspce ((𝐴𝐵𝜓) → ∃𝑥𝐵 𝜑)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem rspce
StepHypRef Expression
1 nfcv 2258 . . . 4 𝑥𝐴
2 nfv 1493 . . . . 5 𝑥 𝐴𝐵
3 rspc.1 . . . . 5 𝑥𝜓
42, 3nfan 1529 . . . 4 𝑥(𝐴𝐵𝜓)
5 eleq1 2180 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
6 rspc.2 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
75, 6anbi12d 464 . . . 4 (𝑥 = 𝐴 → ((𝑥𝐵𝜑) ↔ (𝐴𝐵𝜓)))
81, 4, 7spcegf 2743 . . 3 (𝐴𝐵 → ((𝐴𝐵𝜓) → ∃𝑥(𝑥𝐵𝜑)))
98anabsi5 553 . 2 ((𝐴𝐵𝜓) → ∃𝑥(𝑥𝐵𝜑))
10 df-rex 2399 . 2 (∃𝑥𝐵 𝜑 ↔ ∃𝑥(𝑥𝐵𝜑))
119, 10sylibr 133 1 ((𝐴𝐵𝜓) → ∃𝑥𝐵 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1316  wnf 1421  wex 1453  wcel 1465  wrex 2394
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rex 2399  df-v 2662
This theorem is referenced by:  rspcev  2763  bezoutlemmain  11613
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