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Theorem drsb1 2074
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 1737 . . . . 5  |-  ( x  =  y  ->  (
x  =  z  <->  y  =  z ) )
21sps 1801 . . . 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 2025 . . 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 1702 . 2  |-  ( [ z  /  x ] ph 
<->  ( ( x  =  z  ->  ph )  /\  E. x ( x  =  z  /\  ph )
) )
8 df-sb 1702 . 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 1368   E.wex 1587   [wsb 1701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-10 1776  ax-12 1793  ax-13 1944
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1588  df-nf 1591  df-sb 1702
This theorem is referenced by:  sbco3  2120  sbco3OLD  2121  sb9iOLD  2134  iotaeq  5473
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