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Theorem sbhypf 2649
Description: Introduce an explicit substitution into an implicit substitution hypothesis. See also csbhypf . (Contributed by Raph Levien, 10-Apr-2004.)
Hypotheses
Ref Expression
sbhypf.1  |-  F/ x ps
sbhypf.2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
sbhypf  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  ps ) )
Distinct variable groups:    x, A    x, y
Allowed substitution hints:    ph( x, y)    ps( x, y)    A( y)

Proof of Theorem sbhypf
StepHypRef Expression
1 vex 2605 . . 3  |-  y  e. 
_V
2 eqeq1 2088 . . 3  |-  ( x  =  y  ->  (
x  =  A  <->  y  =  A ) )
31, 2ceqsexv 2639 . 2  |-  ( E. x ( x  =  y  /\  x  =  A )  <->  y  =  A )
4 nfs1v 1857 . . . 4  |-  F/ x [ y  /  x ] ph
5 sbhypf.1 . . . 4  |-  F/ x ps
64, 5nfbi 1522 . . 3  |-  F/ x
( [ y  /  x ] ph  <->  ps )
7 sbequ12 1695 . . . . 5  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
87bicomd 139 . . . 4  |-  ( x  =  y  ->  ( [ y  /  x ] ph  <->  ph ) )
9 sbhypf.2 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
108, 9sylan9bb 450 . . 3  |-  ( ( x  =  y  /\  x  =  A )  ->  ( [ y  /  x ] ph  <->  ps )
)
116, 10exlimi 1526 . 2  |-  ( E. x ( x  =  y  /\  x  =  A )  ->  ( [ y  /  x ] ph  <->  ps ) )
123, 11sylbir 133 1  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285   F/wnf 1390   E.wex 1422   [wsb 1686
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-v 2604
This theorem is referenced by:  mob2  2773  tfisi  4336  ralxpf  4510  rexxpf  4511  nn0ind-raph  8545
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