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Theorem sbalyz 1737
Description: Move universal quantifier in and out of substitution. Identical to sbal 1738 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 ax-ial 1358 . . . 4 (xφxxφ)
21hbsbv 1683 . . 3 ([z / y]xφx[z / y]xφ)
3 ax-4 1333 . . . . 5 (xφφ)
43sbimi 1526 . . . 4 ([z / y]xφ → [z / y]φ)
54alimi 1275 . . 3 (x[z / y]xφx[z / y]φ)
62, 5syl 13 . 2 ([z / y]xφx[z / y]φ)
7 sb6 1637 . . . . 5 ([z / y]φy(y = zφ))
87albii 1290 . . . 4 (x[z / y]φxy(y = zφ))
9 alcom 1297 . . . 4 (xy(y = zφ) ↔ yx(y = zφ))
108, 9bitri 171 . . 3 (x[z / y]φyx(y = zφ))
11 ax-17 1349 . . . . . 6 (y = zx y = z)
12 alim 1277 . . . . . 6 (x(y = zφ) → (x y = zxφ))
1311, 12syl5 26 . . . . 5 (x(y = zφ) → (y = zxφ))
1413alimi 1275 . . . 4 (yx(y = zφ) → y(y = zxφ))
15 sb2 1529 . . . 4 (y(y = zxφ) → [z / y]xφ)
1614, 15syl 13 . . 3 (yx(y = zφ) → [z / y]xφ)
1710, 16sylbi 112 . 2 (x[z / y]φ → [z / y]xφ)
186, 17impbii 115 1 ([z / y]xφx[z / y]φ)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 96  wal 1266   = wceq 1324  [wsbc 1523
This theorem is referenced by:  sbal  1738
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-5 1267  ax-7 1268  ax-gen 1269  ax-ie1 1314  ax-ie2 1315  ax-8 1328  ax-11 1330  ax-4 1333  ax-17 1349  ax-i9 1353  ax-ial 1358  ax-i5r 1359
This theorem depends on definitions:  df-bi 108  df-sb 1525
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