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Theorem sbco3v 1840
Description: Version of sbco3 1845 with a distinct variable constraint between x and y. (Contributed by Jim Kingdon, 19-Feb-2018.)
Assertion
Ref Expression
sbco3v ([z / y][y / x]φ ↔ [z / x][x / y]φ)
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y,z)

Proof of Theorem sbco3v
StepHypRef Expression
1 nfs1v 1812 . . . 4 x[y / x]φ
21nfri 1409 . . 3 ([y / x]φx[y / x]φ)
32sbco2v 1818 . 2 ([z / x][x / y][y / x]φ ↔ [z / y][y / x]φ)
4 sbco 1839 . . 3 ([x / y][y / x]φ ↔ [x / y]φ)
54sbbii 1645 . 2 ([z / x][x / y][y / x]φ ↔ [z / x][x / y]φ)
63, 5bitr3i 175 1 ([z / y][y / x]φ ↔ [z / x][x / y]φ)
Colors of variables: wff set class
Syntax hints:  wb 98  [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-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-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643
This theorem is referenced by:  sbcomv  1842
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