Theorem sbid 1767

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 1694	. . 3 ⊢ 𝑥 = 𝑥
2		sbequ12 1764	. . 3 ⊢ (𝑥 = 𝑥 → (𝜑 ↔ [𝑥 / 𝑥]𝜑))
3	1, 2	ax-mp 5	. 2 ⊢ (𝜑 ↔ [𝑥 / 𝑥]𝜑)
4	3	bicomi 131	1 ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 104  [wsb 1755

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523

This theorem depends on definitions:  df-bi 116  df-sb 1756

This theorem is referenced by:  abid  2158  sbceq1a  2964  sbcid  2970