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

Theorem nfexd 1641
Description: If x is not free in φ, it is not free in yφ. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof rewritten by Jim Kingdon, 7-Feb-2018.)
Hypotheses
Ref Expression
nfald.1 yφ
nfald.2 (φ → Ⅎxψ)
Assertion
Ref Expression
nfexd (φ → Ⅎxyψ)

Proof of Theorem nfexd
StepHypRef Expression
1 nfald.1 . . . . . . 7 yφ
21nfri 1409 . . . . . 6 (φyφ)
3 nfald.2 . . . . . . 7 (φ → Ⅎxψ)
4 df-nf 1347 . . . . . . 7 (Ⅎxψx(ψxψ))
53, 4sylib 127 . . . . . 6 (φx(ψxψ))
62, 5alrimih 1355 . . . . 5 (φyx(ψxψ))
7 alcom 1364 . . . . 5 (yx(ψxψ) ↔ xy(ψxψ))
86, 7sylib 127 . . . 4 (φxy(ψxψ))
9 exim 1487 . . . . 5 (y(ψxψ) → (yψyxψ))
109alimi 1341 . . . 4 (xy(ψxψ) → x(yψyxψ))
118, 10syl 14 . . 3 (φx(yψyxψ))
12 19.12 1552 . . . . 5 (yxψxyψ)
1312imim2i 12 . . . 4 ((yψyxψ) → (yψxyψ))
1413alimi 1341 . . 3 (x(yψyxψ) → x(yψxyψ))
1511, 14syl 14 . 2 (φx(yψxyψ))
16 df-nf 1347 . 2 (Ⅎxyψx(yψxyψ))
1715, 16sylibr 137 1 (φ → Ⅎxyψ)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1240  wnf 1346  wex 1378
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-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-ial 1424
This theorem depends on definitions:  df-bi 110  df-nf 1347
This theorem is referenced by:  nfsbxy  1815  nfsbxyt  1816  nfeudv  1912  nfmod  1914  nfeld  2190  nfrexdxy  2351
  Copyright terms: Public domain W3C validator