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Theorem drsb1 1887
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, 5-Aug-1993.)
Assertion
Ref Expression
drsb1  |-  ( A. x  x  =  y  ->  ( [ z  /  x ] ph  <->  [ z  /  y ] ph ) )

Proof of Theorem drsb1
StepHypRef Expression
1 equequ1 1829 . . . . 5  |-  ( x  =  y  ->  (
x  =  z  <->  y  =  z ) )
21a4s 1700 . . . 4  |-  ( A. x  x  =  y  ->  ( x  =  z  <-> 
y  =  z ) )
32imbi1d 310 . . 3  |-  ( A. x  x  =  y  ->  ( ( x  =  z  ->  ph )  <->  ( y  =  z  ->  ph )
) )
42anbi1d 688 . . . 4  |-  ( A. x  x  =  y  ->  ( ( x  =  z  /\  ph )  <->  ( y  =  z  /\  ph ) ) )
54drex1 1860 . . 3  |-  ( A. x  x  =  y  ->  ( E. x ( x  =  z  /\  ph )  <->  E. y ( y  =  z  /\  ph ) ) )
63, 5anbi12d 694 . 2  |-  ( A. x  x  =  y  ->  ( ( ( x  =  z  ->  ph )  /\  E. x ( x  =  z  /\  ph ) )  <->  ( (
y  =  z  ->  ph )  /\  E. y
( y  =  z  /\  ph ) ) ) )
7 df-sb 1884 . 2  |-  ( [ z  /  x ] ph 
<->  ( ( x  =  z  ->  ph )  /\  E. x ( x  =  z  /\  ph )
) )
8 df-sb 1884 . 2  |-  ( [ z  /  y ]
ph 
<->  ( ( y  =  z  ->  ph )  /\  E. y ( y  =  z  /\  ph )
) )
96, 7, 83bitr4g 281 1  |-  ( A. x  x  =  y  ->  ( [ z  /  x ] ph  <->  [ z  /  y ] ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360   A.wal 1532   E.wex 1537   [wsb 1883
This theorem is referenced by:  sbequi  1952  nfsb4t  1973  sbco3  1983  sbcom  1984  sb9i  1989  iotaeq  6198
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1538  df-nf 1540  df-sb 1884
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