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 2045	. 2 ⊢ 𝑥 = 𝑥
2		sbequ12r 2289	. 2 ⊢ (𝑥 = 𝑥 → ([𝑥 / 𝑥]𝜑 ↔ 𝜑))
3	1, 2	ax-mp 5	1 ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑)

This proof depends on syntax axioms:   ↔ wb 209  [wsb 2099

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-12 2215

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2100

This theorem is used by:  sbco  2538  sbidm  2541  abid  2744  sbceq1a  3753  sbcid  3759  frege58bid  44729  ichid  48338  sbidd  50631  sbidd-misc  50632