Theorem sbid 2099

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 1925	. 2 ⊢ 𝑥 = 𝑥
2		sbequ12r 2097	. 2 ⊢ (𝑥 = 𝑥 → ([𝑥 / 𝑥]𝜑 ↔ 𝜑))
3	1, 2	ax-mp 5	1 ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 194  [wsb 1866

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-12 2033

This theorem depends on definitions:  df-bi 195  df-an 384  df-ex 1695  df-sb 1867

This theorem is referenced by:  sbco  2399  sbidm  2401  sbal2  2448  abid  2597  sbceq1a  3412  sbcid  3418  frege58bid  37012  sbidd  42214  sbidd-misc  42215