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Theorem sbhypf 2833
Description: Introduce an explicit substitution into an implicit substitution hypothesis. See also csbhypf 3116. (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 2791 . . 3  |-  y  e. 
_V
2 eqeq1 2289 . . 3  |-  ( x  =  y  ->  (
x  =  A  <->  y  =  A ) )
31, 2ceqsexv 2823 . 2  |-  ( E. x ( x  =  y  /\  x  =  A )  <->  y  =  A )
4 nfs1v 2045 . . . 4  |-  F/ x [ y  /  x ] ph
5 sbhypf.1 . . . 4  |-  F/ x ps
64, 5nfbi 1772 . . 3  |-  F/ x
( [ y  /  x ] ph  <->  ps )
7 sbequ12 1860 . . . . 5  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
87bicomd 192 . . . 4  |-  ( x  =  y  ->  ( [ y  /  x ] ph  <->  ph ) )
9 sbhypf.2 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
108, 9sylan9bb 680 . . 3  |-  ( ( x  =  y  /\  x  =  A )  ->  ( [ y  /  x ] ph  <->  ps )
)
116, 10exlimi 1801 . 2  |-  ( E. x ( x  =  y  /\  x  =  A )  ->  ( [ y  /  x ] ph  <->  ps ) )
123, 11sylbir 204 1  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1528   F/wnf 1531    = wceq 1623   [wsb 1629
This theorem is referenced by:  mob2  2945  ralxpf  4830  ac6sf  8116  nn0ind-raph  10112  ac6gf  26411  fdc1  26456
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-v 2790
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