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Theorem sblbis 2146
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 2145 . 2  |-  ( [ y  /  x ]
( ch  <->  ph )  <->  ( [
y  /  x ] ch 
<->  [ y  /  x ] ph ) )
2 sblbis.1 . . 3  |-  ( [ y  /  x ] ph 
<->  ps )
32bibi2i 305 . 2  |-  ( ( [ y  /  x ] ch  <->  [ y  /  x ] ph )  <->  ( [
y  /  x ] ch 
<->  ps ) )
41, 3bitri 241 1  |-  ( [ y  /  x ]
( ch  <->  ph )  <->  ( [
y  /  x ] ch 
<->  ps ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177   [wsb 1658
This theorem is referenced by:  sb8eu  2298  sb8iota  5417
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659
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