Theorem sbco 1463

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 1400	. . 3 |- [y / x]y = x
2		sbequ12 1383	. . . . 5 |- (y = x -> (ph <-> [x / y]ph))
3	2	bicomd 577	. . . 4 |- (y = x -> ([x / y]ph <-> ph))
4	3	sbimi 1375	. . 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 1447	. 2 |- ([y / x]([x / y]ph <-> ph) <-> ([y / x][x / y]ph <-> [y / x]ph))
7	5, 6	mpbi 205	1 |- ([y / x][x / y]ph <-> [y / x]ph)

Syntax hints:   <-> wb 162  [wsbc 1372

This theorem is referenced by:  sbid2 1464  sbco3 1469  sb6rfOLD 1474  sb9i 1478

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1142  ax-gen 1143  ax-8 1144  ax-10 1146  ax-12 1148  ax-4 1157  ax-5o 1159  ax-6o 1162  ax-9o 1319  ax-10o 1338  ax-11o 1426

This theorem depends on definitions:  df-bi 163  df-an 241  df-ex 1165  df-sb 1374

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