Theorem sbid 1697

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 1629	. . 3 ⊢ 𝑥 = 𝑥
2		sbequ12 1694	. . 3 ⊢ (𝑥 = 𝑥 → (𝜑 ↔ [𝑥 / 𝑥]𝜑))
3	1, 2	ax-mp 7	. 2 ⊢ (𝜑 ↔ [𝑥 / 𝑥]𝜑)
4	3	bicomi 130	1 ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 103  [wsb 1685

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-4 1440  ax-17 1459  ax-i9 1463

This theorem depends on definitions:  df-bi 115  df-sb 1686

This theorem is referenced by:  abid  2069  sbceq1a  2824  sbcid  2830