Theorem sbid 2292

Description: An identity theorem for substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by NM, 26-May-1993.) (Proof shortened by Wolf Lammen, 30-Sep-2018.)

Assertion

Ref	Expression
sbid	⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑)

Proof of Theorem sbid

Step	Hyp	Ref	Expression
1		equid 2034	. 2 ⊢ 𝑥 = 𝑥
2		sbequ12r 2289	. 2 ⊢ (𝑥 = 𝑥 → ([𝑥 / 𝑥]𝜑 ↔ 𝜑))
3	1, 2	ax-mp 5	1 ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 208  [wsb 2092

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-12 2214

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1802  df-sb 2093

This theorem is referenced by:  sbcovOLD  2294  sbco  2540  sbidm  2543  abid  2746  sbceq1a  3757  sbcid  3763  frege58bid  44483  ichid  48062  sbidd  50344  sbidd-misc  50345