Users' Mathboxes Mathbox for Giovanni Mascellani < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sbeqi Structured version   Unicode version

Theorem sbeqi 32136
Description: Equality deduction for substitution. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
Assertion
Ref Expression
sbeqi  |-  ( ( x  =  y  /\  A. z ( ph  <->  ps )
)  ->  ( [
x  /  z ]
ph 
<->  [ y  /  z ] ps ) )

Proof of Theorem sbeqi
StepHypRef Expression
1 spsbbi 2194 . 2  |-  ( A. z ( ph  <->  ps )  ->  ( [ x  / 
z ] ph  <->  [ x  /  z ] ps ) )
2 sbequ 2168 . 2  |-  ( x  =  y  ->  ( [ x  /  z ] ps  <->  [ y  /  z ] ps ) )
31, 2sylan9bbr 705 1  |-  ( ( x  =  y  /\  A. z ( ph  <->  ps )
)  ->  ( [
x  /  z ]
ph 
<->  [ y  /  z ] ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370   A.wal 1435    = wceq 1437   [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 1838  ax-10 1886  ax-12 1904  ax-13 2052
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: (None)
  Copyright terms: Public domain W3C validator