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

Theorem spsbbi 2196
Description: Specialization of biconditional. (Contributed by NM, 2-Jun-1993.)
Assertion
Ref Expression
spsbbi  |-  ( A. x ( ph  <->  ps )  ->  ( [ y  /  x ] ph  <->  [ y  /  x ] ps )
)

Proof of Theorem spsbbi
StepHypRef Expression
1 stdpc4 2147 . 2  |-  ( A. x ( ph  <->  ps )  ->  [ y  /  x ] ( ph  <->  ps )
)
2 sbbi 2195 . 2  |-  ( [ y  /  x ]
( ph  <->  ps )  <->  ( [
y  /  x ] ph 
<->  [ y  /  x ] ps ) )
31, 2sylib 199 1  |-  ( A. x ( ph  <->  ps )  ->  ( [ y  /  x ] ph  <->  [ y  /  x ] ps )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187   A.wal 1435   [wsb 1786
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 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-12 1905  ax-13 2053
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-ex 1660  df-nf 1664  df-sb 1787
This theorem is referenced by:  sbbid  2197  sbeqi  32317
  Copyright terms: Public domain W3C validator