Theorem sbcom 1963

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 1962	. 2 |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] [ x / z ] ph )
2		sbcocom 1958	. 2 |- ( [ y / z ] [ z / x ] ph <-> [ y / z ] [ y / x ] ph )
3		sbcocom 1958	. 2 |- ( [ y / x ] [ x / z ] ph <-> [ y / x ] [ y / z ] ph )
4	1, 2, 3	3bitr3i 209	1 |- ( [ y / z ] [ y / x ] ph <-> [ y / x ] [ y / z ] ph )

Syntax hints:   <-> wb 104   [wsb 1750

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751

This theorem is referenced by:  sb9  1967