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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]𝜑𝜑)

Proof of Theorem sbid
StepHypRef Expression
1 equid 1969 . 2 𝑥 = 𝑥
2 sbequ12r 2180 . 2 (𝑥 = 𝑥 → ([𝑥 / 𝑥]𝜑𝜑))
31, 2ax-mp 5 1 ([𝑥 / 𝑥]𝜑𝜑)
Colors of variables: wff setvar class
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
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