Theorem sbco 1247

Description: A composition law for substitution.

Assertion

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

Proof of Theorem sbco

Step	Hyp	Ref	Expression
1		equsb2 1190	. . 3 |- [y / x]y = x
2		sbequ12 1177	. . . . 5 |- (y = x -> (ph <-> [x / y]ph))
3	2	bicomd 519	. . . 4 |- (y = x -> ([x / y]ph <-> ph))
4	3	sbimi 1169	. . 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 1234	. 2 |- ([y / x]([x / y]ph <-> ph) <-> ([y / x][x / y]ph <-> [y / x]ph))
7	5, 6	mpbi 189	1 |- ([y / x][x / y]ph <-> [y / x]ph)

Syntax hints:   <-> wb 146  [wsbc 1166

This theorem is referenced by:  sbid2 1248  sbco3 1252  sb6rf 1255  sb9i 1258

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-11o 1213

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-sb 1168

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