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

Theorem sbalyz 1792
Description: Move universal quantifier in and out of substitution. Identical to sbal 1793 except that it has an additional distinct variable constraint on y and z. (Contributed by Jim Kingdon, 29-Dec-2017.)
Assertion
Ref Expression
sbalyz ([z / y]xφx[z / y]φ)
Distinct variable group:   x,y,z
Allowed substitution hints:   φ(x,y,z)

Proof of Theorem sbalyz
StepHypRef Expression
1 nfa1 1373 . . . 4 xxφ
21nfsbxy 1735 . . 3 x[z / y]xφ
3 ax-4 1339 . . . 4 (xφφ)
43sbimi 1569 . . 3 ([z / y]xφ → [z / y]φ)
52, 4alrimi 1353 . 2 ([z / y]xφx[z / y]φ)
6 sb6 1686 . . . . 5 ([z / y]φy(y = zφ))
76albii 1296 . . . 4 (x[z / y]φxy(y = zφ))
8 alcom 1303 . . . 4 (xy(y = zφ) ↔ yx(y = zφ))
97, 8bitri 171 . . 3 (x[z / y]φyx(y = zφ))
10 nfv 1359 . . . . . 6 x y = z
1110stdpc5 1413 . . . . 5 (x(y = zφ) → (y = zxφ))
1211alimi 1281 . . . 4 (yx(y = zφ) → y(y = zxφ))
13 sb2 1572 . . . 4 (y(y = zxφ) → [z / y]xφ)
1412, 13syl 13 . . 3 (yx(y = zφ) → [z / y]xφ)
159, 14sylbi 112 . 2 (x[z / y]φ → [z / y]xφ)
165, 15impbii 115 1 ([z / y]xφx[z / y]φ)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 96  wal 1272  [wsb 1567
This theorem is referenced by:  sbal  1793
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 97  ax-ia2 98  ax-ia3 99  ax-io 610  ax-5 1273  ax-7 1274  ax-gen 1275  ax-ie1 1320  ax-ie2 1321  ax-8 1334  ax-10 1335  ax-11 1336  ax-i12 1337  ax-bnd 1338  ax-4 1339  ax-17 1357  ax-i9 1361  ax-ial 1366  ax-i5r 1367
This theorem depends on definitions:  df-bi 108  df-nf 1287  df-sb 1568
  Copyright terms: Public domain W3C validator