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

Theorem spcimedv 2798
Description: Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
spcimdv.1 (𝜑𝐴𝐵)
spcimedv.2 ((𝜑𝑥 = 𝐴) → (𝜒𝜓))
Assertion
Ref Expression
spcimedv (𝜑 → (𝜒 → ∃𝑥𝜓))
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥   𝜒,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝐵(𝑥)

Proof of Theorem spcimedv
StepHypRef Expression
1 spcimedv.2 . . . 4 ((𝜑𝑥 = 𝐴) → (𝜒𝜓))
21ex 114 . . 3 (𝜑 → (𝑥 = 𝐴 → (𝜒𝜓)))
32alrimiv 1854 . 2 (𝜑 → ∀𝑥(𝑥 = 𝐴 → (𝜒𝜓)))
4 spcimdv.1 . 2 (𝜑𝐴𝐵)
5 nfv 1508 . . 3 𝑥𝜒
6 nfcv 2299 . . 3 𝑥𝐴
75, 6spcimegft 2790 . 2 (∀𝑥(𝑥 = 𝐴 → (𝜒𝜓)) → (𝐴𝐵 → (𝜒 → ∃𝑥𝜓)))
83, 4, 7sylc 62 1 (𝜑 → (𝜒 → ∃𝑥𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wal 1333   = wceq 1335  wex 1472  wcel 2128
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714
This theorem is referenced by:  rspcimedv  2818  fihashf1rn  10667
  Copyright terms: Public domain W3C validator