ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  rspce GIF version

Theorem rspce 2756
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 2256 . . . 4 𝑥𝐴
2 nfv 1491 . . . . 5 𝑥 𝐴𝐵
3 rspc.1 . . . . 5 𝑥𝜓
42, 3nfan 1527 . . . 4 𝑥(𝐴𝐵𝜓)
5 eleq1 2178 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
6 rspc.2 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
75, 6anbi12d 462 . . . 4 (𝑥 = 𝐴 → ((𝑥𝐵𝜑) ↔ (𝐴𝐵𝜓)))
81, 4, 7spcegf 2741 . . 3 (𝐴𝐵 → ((𝐴𝐵𝜓) → ∃𝑥(𝑥𝐵𝜑)))
98anabsi5 551 . 2 ((𝐴𝐵𝜓) → ∃𝑥(𝑥𝐵𝜑))
10 df-rex 2397 . 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 1314  wnf 1419  wex 1451  wcel 1463  wrex 2392
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-rex 2397  df-v 2660
This theorem is referenced by:  rspcev  2761  bezoutlemmain  11582
  Copyright terms: Public domain W3C validator