MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbhypf Unicode version

Theorem sbhypf 2846
Description: Introduce an explicit substitution into an implicit substitution hypothesis. See also csbhypf 3129. (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 2804 . . 3  |-  y  e. 
_V
2 eqeq1 2302 . . 3  |-  ( x  =  y  ->  (
x  =  A  <->  y  =  A ) )
31, 2ceqsexv 2836 . 2  |-  ( E. x ( x  =  y  /\  x  =  A )  <->  y  =  A )
4 nfs1v 2058 . . . 4  |-  F/ x [ y  /  x ] ph
5 sbhypf.1 . . . 4  |-  F/ x ps
64, 5nfbi 1784 . . 3  |-  F/ x
( [ y  /  x ] ph  <->  ps )
7 sbequ12 1872 . . . . 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 1813 . 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 1531   F/wnf 1534    = wceq 1632   [wsb 1638
This theorem is referenced by:  mob2  2958  ralxpf  4846  ac6sf  8132  nn0ind-raph  10128  ac6gf  26514  fdc1  26559
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-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-v 2803
  Copyright terms: Public domain W3C validator