Theorem sbco 1968

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 1786	. . 3 |- [ y / x ] y = x
2		sbequ12 1771	. . . . 5 |- ( y = x -> ( ph <-> [ x / y ] ph ) )
3	2	bicomd 141	. . . 4 |- ( y = x -> ( [ x / y ] ph <-> ph ) )
4	3	sbimi 1764	. . 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 1959	. 2 |- ( [ y / x ] ( [ x / y ] ph <-> ph ) <-> ( [ y / x ] [ x / y ] ph <-> [ y / x ] ph ) )
7	5, 6	mpbi 145	1 |- ( [ y / x ] [ x / y ] ph <-> [ y / x ] ph )

Syntax hints:   <-> wb 105   [wsb 1762

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763

This theorem is referenced by:  sbco3v  1969