Theorem sbid 2252

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 2015	. 2 ⊢ 𝑥 = 𝑥
2		sbequ12r 2249	. 2 ⊢ (𝑥 = 𝑥 → ([𝑥 / 𝑥]𝜑 ↔ 𝜑))
3	1, 2	ax-mp 5	1 ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 208  [wsb 2065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-12 2172

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1777  df-sb 2066

This theorem is referenced by:  sbcov  2253  sbco  2545  sbidm  2548  sbal2OLD  2570  abid  2803  sbceq1a  3783  sbcid  3789  frege58bid  40241  ichid  43604  sbidd  44810  sbidd-misc  44811