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

Theorem sbhypf 3128
Description: Introduce an explicit substitution into an implicit substitution hypothesis. See also csbhypf 3414. (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 3083 . . 3  |-  y  e. 
_V
2 eqeq1 2426 . . 3  |-  ( x  =  y  ->  (
x  =  A  <->  y  =  A ) )
31, 2ceqsexv 3118 . 2  |-  ( E. x ( x  =  y  /\  x  =  A )  <->  y  =  A )
4 nfs1v 2236 . . . 4  |-  F/ x [ y  /  x ] ph
5 sbhypf.1 . . . 4  |-  F/ x ps
64, 5nfbi 1994 . . 3  |-  F/ x
( [ y  /  x ] ph  <->  ps )
7 sbequ12 2051 . . . . 5  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
87bicomd 204 . . . 4  |-  ( x  =  y  ->  ( [ y  /  x ] ph  <->  ph ) )
9 sbhypf.2 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
108, 9sylan9bb 704 . . 3  |-  ( ( x  =  y  /\  x  =  A )  ->  ( [ y  /  x ] ph  <->  ps )
)
116, 10exlimi 1972 . 2  |-  ( E. x ( x  =  y  /\  x  =  A )  ->  ( [ y  /  x ] ph  <->  ps ) )
123, 11sylbir 216 1  |-  ( y  =  A  ->  ( [ y  /  x ] ph  <->  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437   E.wex 1657   F/wnf 1661   [wsb 1790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-v 3082
This theorem is referenced by:  mob2  3250  reu2eqd  3267  cbvmptf  4514  ralxpf  5000  tfisi  6699  ac6sf  8926  nn0ind-raph  11042  ac6sf2  28228  nn0min  28391  ac6gf  32023  fdc1  32039
  Copyright terms: Public domain W3C validator