Theorem sbcom 1891

Description: A commutativity law for substitution. (Contributed by NM, 27-May-1997.) (Proof rewritten by Jim Kingdon, 22-Mar-2018.)

Assertion

Ref	Expression
sbcom	|- ( [ y / z ] [ y / x ] ph <-> [ y / x ] [ y / z ] ph )

Proof of Theorem sbcom

Step	Hyp	Ref	Expression
1		sbco3 1890	. 2 |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] [ x / z ] ph )
2		sbcocom 1886	. 2 |- ( [ y / z ] [ z / x ] ph <-> [ y / z ] [ y / x ] ph )
3		sbcocom 1886	. 2 |- ( [ y / x ] [ x / z ] ph <-> [ y / x ] [ y / z ] ph )
4	1, 2, 3	3bitr3i 208	1 |- ( [ y / z ] [ y / x ] ph <-> [ y / x ] [ y / z ] ph )

Syntax hints:   <-> wb 103   [wsb 1686

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469

This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687

This theorem is referenced by:  sb9  1897