Theorem sbid 1715

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 1645	. . 3 ⊢ 𝑥 = 𝑥
2		sbequ12 1712	. . 3 ⊢ (𝑥 = 𝑥 → (𝜑 ↔ [𝑥 / 𝑥]𝜑))
3	1, 2	ax-mp 7	. 2 ⊢ (𝜑 ↔ [𝑥 / 𝑥]𝜑)
4	3	bicomi 131	1 ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 104  [wsb 1703

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-4 1455  ax-17 1474  ax-i9 1478

This theorem depends on definitions:  df-bi 116  df-sb 1704

This theorem is referenced by:  abid  2088  sbceq1a  2871  sbcid  2877