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Theorem bnj610 29092
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 2804 . . . 4  |-  y  e. 
_V
2 bnj610.3 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  ps' ) )
31, 2sbcie 3038 . . 3  |-  ( [. y  /  x ]. ph  <->  ps' )
43sbcbii 3059 . 2  |-  ( [. A  /  y ]. [. y  /  x ]. ph  <->  [. A  / 
y ]. ps' )
5 sbcco 3026 . 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 3038 . 2  |-  ( [. A  /  y ]. ps'  <->  ps )
94, 5, 83bitr3i 266 1  |-  ( [. A  /  x ]. ph  <->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1632    e. wcel 1696   _Vcvv 2801   [.wsbc 3004
This theorem is referenced by:  bnj611  29266  bnj1000  29289
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-sbc 3005
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