MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  drsb1 Structured version   Unicode version

Theorem drsb1 2084
Description: Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 2-Jun-1993.)
Assertion
Ref Expression
drsb1  |-  ( A. x  x  =  y  ->  ( [ z  /  x ] ph  <->  [ z  /  y ] ph ) )

Proof of Theorem drsb1
StepHypRef Expression
1 equequ1 1742 . . . . 5  |-  ( x  =  y  ->  (
x  =  z  <->  y  =  z ) )
21sps 1809 . . . 4  |-  ( A. x  x  =  y  ->  ( x  =  z  <-> 
y  =  z ) )
32imbi1d 317 . . 3  |-  ( A. x  x  =  y  ->  ( ( x  =  z  ->  ph )  <->  ( y  =  z  ->  ph )
) )
42anbi1d 704 . . . 4  |-  ( A. x  x  =  y  ->  ( ( x  =  z  /\  ph )  <->  ( y  =  z  /\  ph ) ) )
54drex1 2035 . . 3  |-  ( A. x  x  =  y  ->  ( E. x ( x  =  z  /\  ph )  <->  E. y ( y  =  z  /\  ph ) ) )
63, 5anbi12d 710 . 2  |-  ( A. x  x  =  y  ->  ( ( ( x  =  z  ->  ph )  /\  E. x ( x  =  z  /\  ph ) )  <->  ( (
y  =  z  ->  ph )  /\  E. y
( y  =  z  /\  ph ) ) ) )
7 df-sb 1707 . 2  |-  ( [ z  /  x ] ph 
<->  ( ( x  =  z  ->  ph )  /\  E. x ( x  =  z  /\  ph )
) )
8 df-sb 1707 . 2  |-  ( [ z  /  y ]
ph 
<->  ( ( y  =  z  ->  ph )  /\  E. y ( y  =  z  /\  ph )
) )
96, 7, 83bitr4g 288 1  |-  ( A. x  x  =  y  ->  ( [ z  /  x ] ph  <->  [ z  /  y ] ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1372   E.wex 1591   [wsb 1706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-12 1798  ax-13 1961
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1592  df-nf 1595  df-sb 1707
This theorem is referenced by:  sbco3  2130  sbco3OLD  2131  sb9iOLD  2144  iotaeq  5550
  Copyright terms: Public domain W3C validator