Theorem sbid 1657

Description: An identity theorem for substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by NM, 5-Aug-1993.)

Assertion

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

Proof of Theorem sbid

Step	Hyp	Ref	Expression
1		equid 1589	. . 3 |- x = x
2		sbequ12 1654	. . 3 |- ( x = x -> ( ph <-> [ x / x ] ph ) )
3	1, 2	ax-mp 7	. 2 |- ( ph <-> [ x / x ] ph )
4	3	bicomi 123	1 |- ( [ x / x ] ph <-> ph )

Syntax hints:   <-> wb 98   [wsb 1645

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423

This theorem depends on definitions:  df-bi 110  df-sb 1646

This theorem is referenced by:  abid  2028  sbceq1a  2773  sbcid  2779