Theorem sbid 1167

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

Assertion

Ref	Expression
sbid	⊢ ([x / x]φ ↔ φ)

Proof of Theorem sbid

Step	Hyp	Ref	Expression
1		equid 1113	. . 3 ⊢ x = x
2		sbequ12 1164	. . 3 ⊢ (x = x → (φ ↔ [x / x]φ))
3	1, 2	ax-mp 7	. 2 ⊢ (φ ↔ [x / x]φ)
4	3	bicomi 172	1 ⊢ ([x / x]φ ↔ φ)

Syntax hints:   ↔ wb 146  [wsbc 1153

This theorem is referenced by:  abid 1442  sbceq1a 1915  csbid 1976

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-gen 955  ax-9 1102  ax-12 1104

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 957  df-sb 1155

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