MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbcid Unicode version

Theorem sbcid 3020
Description: An identity theorem for substitution. See sbid 1875. (Contributed by Mario Carneiro, 18-Feb-2017.)
Assertion
Ref Expression
sbcid  |-  ( [. x  /  x ]. ph  <->  ph )

Proof of Theorem sbcid
StepHypRef Expression
1 sbsbc 3008 . 2  |-  ( [ x  /  x ] ph 
<-> 
[. x  /  x ]. ph )
2 sbid 1875 . 2  |-  ( [ x  /  x ] ph 
<-> 
ph )
31, 2bitr3i 242 1  |-  ( [. x  /  x ]. ph  <->  ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   [wsb 1638   [.wsbc 3004
This theorem is referenced by:  ex-natded9.26  20822
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-11 1727  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-sbc 3005
  Copyright terms: Public domain W3C validator