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Theorem equs5 1757
Description: Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
equs5  |-  ( -. 
A. x  x  =  y  ->  ( E. x ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  ph )
) )

Proof of Theorem equs5
StepHypRef Expression
1 hbnae 1656 . 2  |-  ( -. 
A. x  x  =  y  ->  A. x  -.  A. x  x  =  y )
2 hba1 1478 . 2  |-  ( A. x ( x  =  y  ->  ph )  ->  A. x A. x ( x  =  y  ->  ph ) )
3 ax11o 1750 . . 3  |-  ( -. 
A. x  x  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) ) )
43impd 251 . 2  |-  ( -. 
A. x  x  =  y  ->  ( (
x  =  y  /\  ph )  ->  A. x
( x  =  y  ->  ph ) ) )
51, 2, 4exlimdh 1532 1  |-  ( -. 
A. x  x  =  y  ->  ( E. x ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  ph )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102   A.wal 1287   E.wex 1426
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-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693
This theorem is referenced by:  sb3  1759  sb4  1760
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