Theorem sbid 1922

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]φ ↔ φ)

Proof of Theorem sbid

Step	Hyp	Ref	Expression
1		equid 1676	. . 3 ⊢ x = x
2		sbequ12 1919	. . 3 ⊢ (x = x → (φ ↔ [x / x]φ))
3	1, 2	ax-mp 5	. 2 ⊢ (φ ↔ [x / x]φ)
4	3	bicomi 193	1 ⊢ ([x / x]φ ↔ φ)

Syntax hints:   ↔ wb 176  [wsb 1648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-sb 1649

This theorem is referenced by:  abid  2341  sbceq1a  3057  sbcid  3063  csbid  3144