Theorem sbco 1883

Description: A composition law for substitution. (Contributed by NM, 5-Aug-1993.)

Assertion

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

Proof of Theorem sbco

Step	Hyp	Ref	Expression
1		equsb2 1709	. . 3 |- [ y / x ] y = x
2		sbequ12 1694	. . . . 5 |- ( y = x -> ( ph <-> [ x / y ] ph ) )
3	2	bicomd 139	. . . 4 |- ( y = x -> ( [ x / y ] ph <-> ph ) )
4	3	sbimi 1687	. . 3 |- ( [ y / x ] y = x -> [ y / x ] ( [ x / y ] ph <-> ph ) )
5	1, 4	ax-mp 7	. 2 |- [ y / x ] ( [ x / y ] ph <-> ph )
6		sbbi 1874	. 2 |- ( [ y / x ] ( [ x / y ] ph <-> ph ) <-> ( [ y / x ] [ x / y ] ph <-> [ y / x ] ph ) )
7	5, 6	mpbi 143	1 |- ( [ y / x ] [ x / y ] ph <-> [ y / x ] ph )

Syntax hints:   <-> wb 103   [wsb 1685

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686

This theorem is referenced by:  sbco3v  1884