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Theorem drsb1 1721
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 1639 . . . . 5  |-  ( x  =  y  ->  (
x  =  z  <->  y  =  z ) )
21sps 1471 . . . 4  |-  ( A. x  x  =  y  ->  ( x  =  z  <-> 
y  =  z ) )
32imbi1d 229 . . 3  |-  ( A. x  x  =  y  ->  ( ( x  =  z  ->  ph )  <->  ( y  =  z  ->  ph )
) )
42anbi1d 453 . . . 4  |-  ( A. x  x  =  y  ->  ( ( x  =  z  /\  ph )  <->  ( y  =  z  /\  ph ) ) )
54drex1 1720 . . 3  |-  ( A. x  x  =  y  ->  ( E. x ( x  =  z  /\  ph )  <->  E. y ( y  =  z  /\  ph ) ) )
63, 5anbi12d 457 . 2  |-  ( A. x  x  =  y  ->  ( ( ( x  =  z  ->  ph )  /\  E. x ( x  =  z  /\  ph ) )  <->  ( (
y  =  z  ->  ph )  /\  E. y
( y  =  z  /\  ph ) ) ) )
7 df-sb 1687 . 2  |-  ( [ z  /  x ] ph 
<->  ( ( x  =  z  ->  ph )  /\  E. x ( x  =  z  /\  ph )
) )
8 df-sb 1687 . 2  |-  ( [ z  /  y ]
ph 
<->  ( ( y  =  z  ->  ph )  /\  E. y ( y  =  z  /\  ph )
) )
96, 7, 83bitr4g 221 1  |-  ( A. x  x  =  y  ->  ( [ z  /  x ] ph  <->  [ z  /  y ] ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1283   E.wex 1422   [wsb 1686
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-sb 1687
This theorem is referenced by:  sbequi  1761  nfsbxy  1860  nfsbxyt  1861  sbcomxyyz  1888  iotaeq  4905
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