Theorem sbid 868

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		eqid 810	. . 3 ⊢ x = x
2		sbequ12 865	. . 3 ⊢ (x = x → (φ ↔ [x / x]φ))
3	1, 2	ax-mp 6	. 2 ⊢ (φ ↔ [x / x]φ)
4	3	bicomi 150	1 ⊢ ([x / x]φ ↔ φ)

Syntax hints:   ↔ wb 127   = weq 797  [wsb 852

This theorem is referenced by:  abid 1094  sbceq1 1443

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-gen 677  ax-9 799  ax-12 802

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853

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