Theorem sbid 2298

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 2040	. 2 ⊢ 𝑥 = 𝑥
2		sbequ12r 2295	. 2 ⊢ (𝑥 = 𝑥 → ([𝑥 / 𝑥]𝜑 ↔ 𝜑))
3	1, 2	ax-mp 5	1 ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 209  [wsb 2098

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-12 2220

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1808  df-sb 2099

This theorem is referenced by:  sbcovOLD  2300  sbco  2546  sbidm  2549  abid  2752  sbceq1a  3763  sbcid  3769  frege58bid  44580  ichid  48149  sbidd  50445  sbidd-misc  50446