MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  stdpc7 Unicode version

Theorem stdpc7 1858
Description: One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 1650.) Translated to traditional notation, it can be read: " x  =  y  ->  ( ph (
x ,  x )  ->  ph ( x ,  y ) ), provided that  y is free for  x in  ph ( x ,  x
)." Axiom 7 of [Mendelson] p. 95. (Contributed by NM, 15-Feb-2005.)
Assertion
Ref Expression
stdpc7  |-  ( x  =  y  ->  ( [ x  /  y ] ph  ->  ph ) )

Proof of Theorem stdpc7
StepHypRef Expression
1 sbequ2 1631 . 2  |-  ( y  =  x  ->  ( [ x  /  y ] ph  ->  ph ) )
21equcoms 1651 1  |-  ( x  =  y  ->  ( [ x  /  y ] ph  ->  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   [wsb 1629
This theorem is referenced by:  ax16ALT2  1988  sbequi  1999  sb5rf  2030  sb8  2032
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
This theorem depends on definitions:  df-bi 177  df-an 360  df-sb 1630
  Copyright terms: Public domain W3C validator