Theorem sbid 1549

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 1484	. . 3 |- x = x
2		sbequ12 1545	. . 3 |- (x = x -> (ph <-> [x / x]ph))
3	1, 2	ax-mp 7	. 2 |- (ph <-> [x / x]ph)
4	3	bicomi 189	1 |- ([x / x]ph <-> ph)

Syntax hints:   <-> wb 163  [wsbc 1534

This theorem is referenced by:  abid 1873  sbceq1a 2456  csbid 2545  sbssOLD 2981  tratrb 5831  bnj605 13292  bnj606 13293  tratrbVD 16685

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1305  ax-12 1310  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481

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

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