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

Theorem dvelimdf 1889
Description: Deduction form of dvelimf 1888. This version may be useful if we want to avoid ax-17 1416 and use ax-16 1692 instead. (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 11-May-2018.)
Hypotheses
Ref Expression
dvelimdf.1 xφ
dvelimdf.2 zφ
dvelimdf.3 (φ → Ⅎxψ)
dvelimdf.4 (φ → Ⅎzχ)
dvelimdf.5 (φ → (z = y → (ψχ)))
Assertion
Ref Expression
dvelimdf (φ → (¬ x x = y → Ⅎxχ))

Proof of Theorem dvelimdf
StepHypRef Expression
1 dvelimdf.2 . . . 4 zφ
2 dvelimdf.3 . . . 4 (φ → Ⅎxψ)
31, 2alrimi 1412 . . 3 (φzxψ)
4 nfsb4t 1887 . . 3 (zxψ → (¬ x x = y → Ⅎx[y / z]ψ))
53, 4syl 14 . 2 (φ → (¬ x x = y → Ⅎx[y / z]ψ))
6 dvelimdf.1 . . 3 xφ
7 dvelimdf.4 . . . 4 (φ → Ⅎzχ)
8 dvelimdf.5 . . . 4 (φ → (z = y → (ψχ)))
91, 7, 8sbied 1668 . . 3 (φ → ([y / z]ψχ))
106, 9nfbidf 1429 . 2 (φ → (Ⅎx[y / z]ψ ↔ Ⅎxχ))
115, 10sylibd 138 1 (φ → (¬ x x = y → Ⅎxχ))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 98  wal 1240  wnf 1346  [wsb 1642
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-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643
This theorem is referenced by:  dvelimdc  2194
  Copyright terms: Public domain W3C validator