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Theorem sb9v 1851
Description: Like sb9 1852 but with a distinct variable constraint between x and y. (Contributed by Jim Kingdon, 28-Feb-2018.)
Assertion
Ref Expression
sb9v (x[x / y]φy[y / x]φ)
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)

Proof of Theorem sb9v
StepHypRef Expression
1 hbs1 1811 . 2 ([x / y]φy[x / y]φ)
2 hbs1 1811 . 2 ([y / x]φx[y / x]φ)
3 sbequ12 1651 . . . 4 (y = x → (φ ↔ [x / y]φ))
43equcoms 1591 . . 3 (x = y → (φ ↔ [x / y]φ))
5 sbequ12 1651 . . 3 (x = y → (φ ↔ [y / x]φ))
64, 5bitr3d 179 . 2 (x = y → ([x / y]φ ↔ [y / x]φ))
71, 2, 6cbvalh 1633 1 (x[x / y]φy[y / x]φ)
Colors of variables: wff set class
Syntax hints:  wb 98  wal 1240  [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-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643
This theorem is referenced by:  sb9  1852
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