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Theorem sbrbis 1934
Description: Introduce right biconditional inside of a substitution. (Contributed by NM, 18-Aug-1993.)
Hypothesis
Ref Expression
sbrbis.1  |-  ( [ y  /  x ] ph 
<->  ps )
Assertion
Ref Expression
sbrbis  |-  ( [ y  /  x ]
( ph  <->  ch )  <->  ( ps  <->  [ y  /  x ] ch ) )

Proof of Theorem sbrbis
StepHypRef Expression
1 sbbi 1932 . 2  |-  ( [ y  /  x ]
( ph  <->  ch )  <->  ( [
y  /  x ] ph 
<->  [ y  /  x ] ch ) )
2 sbrbis.1 . . 3  |-  ( [ y  /  x ] ph 
<->  ps )
32bibi1i 227 . 2  |-  ( ( [ y  /  x ] ph  <->  [ y  /  x ] ch )  <->  ( ps  <->  [ y  /  x ] ch ) )
41, 3bitri 183 1  |-  ( [ y  /  x ]
( ph  <->  ch )  <->  ( ps  <->  [ y  /  x ] ch ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 104   [wsb 1735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736
This theorem is referenced by:  sbrbif  1935  sbabel  2307
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