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Theorem sblbis 2131
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 2128 . 2  |-  ( [ y  /  x ]
( ch  <->  ph )  <->  ( [
y  /  x ] ch 
<->  [ y  /  x ] ph ) )
2 sblbis.1 . . 3  |-  ( [ y  /  x ] ph 
<->  ps )
32bibi2i 313 . 2  |-  ( ( [ y  /  x ] ch  <->  [ y  /  x ] ph )  <->  ( [
y  /  x ] ch 
<->  ps ) )
41, 3bitri 249 1  |-  ( [ y  /  x ]
( ch  <->  ph )  <->  ( [
y  /  x ] ch 
<->  ps ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184   [wsb 1726
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-12 1840  ax-13 1985
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-ex 1600  df-nf 1604  df-sb 1727
This theorem is referenced by:  sbie  2135  sb8eu  2304  sb8euOLD  2305  sb8iota  5548  wl-sb8eut  29997
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