Theorem sbco 1939

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 1759	. . 3 |- [ y / x ] y = x
2		sbequ12 1744	. . . . 5 |- ( y = x -> ( ph <-> [ x / y ] ph ) )
3	2	bicomd 140	. . . 4 |- ( y = x -> ( [ x / y ] ph <-> ph ) )
4	3	sbimi 1737	. . 3 |- ( [ y / x ] y = x -> [ y / x ] ( [ x / y ] ph <-> ph ) )
5	1, 4	ax-mp 5	. 2 |- [ y / x ] ( [ x / y ] ph <-> ph )
6		sbbi 1930	. 2 |- ( [ y / x ] ( [ x / y ] ph <-> ph ) <-> ( [ y / x ] [ x / y ] ph <-> [ y / x ] ph ) )
7	5, 6	mpbi 144	1 |- ( [ y / x ] [ x / y ] ph <-> [ y / x ] ph )

Syntax hints:   <-> wb 104   [wsb 1735

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736

This theorem is referenced by:  sbco3v  1940