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Theorem drsb1 1792
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 1705 . . . . 5  |-  ( x  =  y  ->  (
x  =  z  <->  y  =  z ) )
21sps 1530 . . . 4  |-  ( A. x  x  =  y  ->  ( x  =  z  <-> 
y  =  z ) )
32imbi1d 230 . . 3  |-  ( A. x  x  =  y  ->  ( ( x  =  z  ->  ph )  <->  ( y  =  z  ->  ph )
) )
42anbi1d 462 . . . 4  |-  ( A. x  x  =  y  ->  ( ( x  =  z  /\  ph )  <->  ( y  =  z  /\  ph ) ) )
54drex1 1791 . . 3  |-  ( A. x  x  =  y  ->  ( E. x ( x  =  z  /\  ph )  <->  E. y ( y  =  z  /\  ph ) ) )
63, 5anbi12d 470 . 2  |-  ( A. x  x  =  y  ->  ( ( ( x  =  z  ->  ph )  /\  E. x ( x  =  z  /\  ph ) )  <->  ( (
y  =  z  ->  ph )  /\  E. y
( y  =  z  /\  ph ) ) ) )
7 df-sb 1756 . 2  |-  ( [ z  /  x ] ph 
<->  ( ( x  =  z  ->  ph )  /\  E. x ( x  =  z  /\  ph )
) )
8 df-sb 1756 . 2  |-  ( [ z  /  y ]
ph 
<->  ( ( y  =  z  ->  ph )  /\  E. y ( y  =  z  /\  ph )
) )
96, 7, 83bitr4g 222 1  |-  ( A. x  x  =  y  ->  ( [ z  /  x ] ph  <->  [ z  /  y ] ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   A.wal 1346   E.wex 1485   [wsb 1755
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527
This theorem depends on definitions:  df-bi 116  df-sb 1756
This theorem is referenced by:  sbequi  1832  nfsbxy  1935  nfsbxyt  1936  sbcomxyyz  1965  iotaeq  5168
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