Theorem sbco 2412

Description: A composition law for substitution. (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 21-Sep-2018.)

Assertion

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

Proof of Theorem sbco

Step	Hyp	Ref	Expression
1		sbcom3 2411	. 2 |- ( [ y / x ] [ x / y ] ph <-> [ y / x ] [ y / y ] ph )
2		sbid 2114	. . 3 |- ( [ y / y ] ph <-> ph )
3	2	sbbii 1887	. 2 |- ( [ y / x ] [ y / y ] ph <-> [ y / x ] ph )
4	1, 3	bitri 264	1 |- ( [ y / x ] [ x / y ] ph <-> [ y / x ] 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-12 2047  ax-13 2246

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

This theorem is referenced by:  sbid2  2413  sbco3  2417  sb6a  2448