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Theorem sbid 1698
Description: An identity theorem for substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
sbid ([𝑥 / 𝑥]𝜑𝜑)

Proof of Theorem sbid
StepHypRef Expression
1 equid 1630 . . 3 𝑥 = 𝑥
2 sbequ12 1695 . . 3 (𝑥 = 𝑥 → (𝜑 ↔ [𝑥 / 𝑥]𝜑))
31, 2ax-mp 7 . 2 (𝜑 ↔ [𝑥 / 𝑥]𝜑)
43bicomi 130 1 ([𝑥 / 𝑥]𝜑𝜑)
Colors of variables: wff set class
Syntax hints:  wb 103  [wsb 1686
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-4 1441  ax-17 1460  ax-i9 1464
This theorem depends on definitions:  df-bi 115  df-sb 1687
This theorem is referenced by:  abid  2070  sbceq1a  2825  sbcid  2831
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