Theorem sbid 1828

Description: An identity theorem for substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint).

Assertion

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

Proof of Theorem sbid

Step	Hyp	Ref	Expression
1		equid 1766	. . 3 |- x = x
2		sbequ12 1825	. . 3 |- (x = x -> (ph <-> [x / x]ph))
3	1, 2	ax-mp 7	. 2 |- (ph <-> [x / x]ph)
4	3	bicomi 268	1 |- ([x / x]ph <-> ph)

Syntax hints:   <-> wb 219  [wsbc 1814

This theorem is referenced by:  abid 2130  sbceq1a 2702  csbid 2778  tratrb 6068  bnj605 14063  bnj606 14064  tratrbVD 17519

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1593  ax-12 1598  ax-4 1608  ax-5o 1610  ax-6o 1613  ax-9o 1763

This theorem depends on definitions:  df-bi 220  df-an 339  df-ex 1616  df-sb 1816

Copyright terms: Public domain