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Theorem bnj610 29115
Description: Pass from equality ( x  =  A) to substitution (
[. A  /  x ].) without the distinct variable restriction ($d  A  x). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj610.1  |-  A  e. 
_V
bnj610.2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
bnj610.3  |-  ( x  =  y  ->  ( ph 
<->  ps' ) )
bnj610.4  |-  ( y  =  A  ->  ( ps'  <->  ps ) )
Assertion
Ref Expression
bnj610  |-  ( [. A  /  x ]. ph  <->  ps )
Distinct variable groups:    y, A    ph, y    ps, y    x, ps'    x, y
Allowed substitution hints:    ph( x)    ps( x)    A( x)    ps'( y)

Proof of Theorem bnj610
StepHypRef Expression
1 vex 2959 . . . 4  |-  y  e. 
_V
2 bnj610.3 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  ps' ) )
31, 2sbcie 3195 . . 3  |-  ( [. y  /  x ]. ph  <->  ps' )
43sbcbii 3216 . 2  |-  ( [. A  /  y ]. [. y  /  x ]. ph  <->  [. A  / 
y ]. ps' )
5 sbcco 3183 . 2  |-  ( [. A  /  y ]. [. y  /  x ]. ph  <->  [. A  /  x ]. ph )
6 bnj610.1 . . 3  |-  A  e. 
_V
7 bnj610.4 . . 3  |-  ( y  =  A  ->  ( ps'  <->  ps ) )
86, 7sbcie 3195 . 2  |-  ( [. A  /  y ]. ps'  <->  ps )
94, 5, 83bitr3i 267 1  |-  ( [. A  /  x ]. ph  <->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1652    e. wcel 1725   _Vcvv 2956   [.wsbc 3161
This theorem is referenced by:  bnj611  29289  bnj1000  29312
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  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-sbc 3162
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