Theorem sbcom 2418

Description: A commutativity law for substitution. (Contributed by NM, 27-May-1997.) (Proof shortened by Wolf Lammen, 20-Sep-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 2417	. 2 |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] [ x / z ] ph )
2		sbcom3 2411	. 2 |- ( [ y / z ] [ z / x ] ph <-> [ y / z ] [ y / x ] ph )
3		sbcom3 2411	. 2 |- ( [ y / x ] [ x / z ] ph <-> [ y / x ] [ y / z ] ph )
4	1, 2, 3	3bitr3i 290	1 |- ( [ y / z ] [ y / x ] ph <-> [ y / x ] [ y / z ] ph )

Syntax hints:   <-> wb 196   [wsb 1880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by:  wl-sbcom3  33372