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Theorem sblbis 1953
Description: Introduce left biconditional inside of a substitution. (Contributed by NM, 19-Aug-1993.)
Hypothesis
Ref Expression
sblbis.1  |-  ( [ y  /  x ] ph 
<->  ps )
Assertion
Ref Expression
sblbis  |-  ( [ y  /  x ]
( ch  <->  ph )  <->  ( [
y  /  x ] ch 
<->  ps ) )

Proof of Theorem sblbis
StepHypRef Expression
1 sbbi 1952 . 2  |-  ( [ y  /  x ]
( ch  <->  ph )  <->  ( [
y  /  x ] ch 
<->  [ y  /  x ] ph ) )
2 sblbis.1 . . 3  |-  ( [ y  /  x ] ph 
<->  ps )
32bibi2i 226 . 2  |-  ( ( [ y  /  x ] ch  <->  [ y  /  x ] ph )  <->  ( [
y  /  x ] ch 
<->  ps ) )
41, 3bitri 183 1  |-  ( [ y  /  x ]
( ch  <->  ph )  <->  ( [
y  /  x ] ch 
<->  ps ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 104   [wsb 1755
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756
This theorem is referenced by:  sb8eu  2032  sb8euh  2042  sb8iota  5167
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