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Theorem bnj610 29345
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 3090 . . . 4  |-  y  e. 
_V
2 bnj610.3 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  ps' ) )
31, 2sbcie 3340 . . 3  |-  ( [. y  /  x ]. ph  <->  ps' )
43sbcbii 3361 . 2  |-  ( [. A  /  y ]. [. y  /  x ]. ph  <->  [. A  / 
y ]. ps' )
5 sbcco 3328 . 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 3340 . 2  |-  ( [. A  /  y ]. ps'  <->  ps )
94, 5, 83bitr3i 278 1  |-  ( [. A  /  x ]. ph  <->  ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    = wceq 1437    e. wcel 1870   _Vcvv 3087   [.wsbc 3305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-v 3089  df-sbc 3306
This theorem is referenced by:  bnj611  29517  bnj1000  29540
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