Theorem sbidm 1627

Description: An idempotent law for substitution. (The proof was shortened by Andrew Salmon, 25-May-2011.)

Assertion

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

Proof of Theorem sbidm

Step	Hyp	Ref	Expression
1		equsb2 1562	. . 3 |- [y / x]y = x
2		sbequ12r 1546	. . . 4 |- (y = x -> ([y / x]ph <-> ph))
3	2	sbimi 1537	. . 3 |- ([y / x]y = x -> [y / x]([y / x]ph <-> ph))
4	1, 3	ax-mp 7	. 2 |- [y / x]([y / x]ph <-> ph)
5		sbbi 1609	. 2 |- ([y / x]([y / x]ph <-> ph) <-> ([y / x][y / x]ph <-> [y / x]ph))
6	4, 5	mpbi 206	1 |- ([y / x][y / x]ph <-> [y / x]ph)

Syntax hints:   <-> wb 163  [wsbc 1534

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-10 1308  ax-12 1310  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-11o 1588

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327  df-sb 1536

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