Theorem sbid 2183

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 1969	. 2 ⊢ 𝑥 = 𝑥
2		sbequ12r 2180	. 2 ⊢ (𝑥 = 𝑥 → ([𝑥 / 𝑥]𝜑 ↔ 𝜑))
3	1, 2	ax-mp 5	1 ⊢ ([𝑥 / 𝑥]𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 198  [wsb 2015

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-12 2106

This theorem depends on definitions:  df-bi 199  df-an 388  df-ex 1743  df-sb 2016

This theorem is referenced by:  sbcov  2184  sbid2vw  2186  sbco  2473  sbidm  2476  sbal2OLD  2497  sbal2OLDOLD  2498  abid  2762  sbceq1a  3692  sbcid  3698  frege58bid  39617  ichid  42987  sbidd  44190  sbidd-misc  44191