Theorem sbid 2261

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 2094	. 2 ⊢ 𝑥 = 𝑥
2		sbequ12r 2259	. 2 ⊢ (𝑥 = 𝑥 → ([𝑥 / 𝑥]𝜑 ↔ 𝜑))
3	1, 2	ax-mp 5	1 ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 196  [wsb 2046

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-12 2196

This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1854  df-sb 2047

This theorem is referenced by:  sbco  2549  sbidm  2551  sbal2  2598  abid  2748  sbceq1a  3587  sbcid  3593  frege58bid  38698  sbidd  42972  sbidd-misc  42973