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Theorem hbex 1745
Description: If  x is not free in  ph, it is not free in  E. y ph. (Contributed by NM, 5-Aug-1993.)
Hypothesis
Ref Expression
hbex.1  |-  ( ph  ->  A. x ph )
Assertion
Ref Expression
hbex  |-  ( E. y ph  ->  A. x E. y ph )

Proof of Theorem hbex
StepHypRef Expression
1 df-ex 1532 . 2  |-  ( E. y ph  <->  -.  A. y  -.  ph )
2 hbex.1 . . . . 5  |-  ( ph  ->  A. x ph )
32hbn 1732 . . . 4  |-  ( -. 
ph  ->  A. x  -.  ph )
43hbal 1722 . . 3  |-  ( A. y  -.  ph  ->  A. x A. y  -.  ph )
54hbn 1732 . 2  |-  ( -. 
A. y  -.  ph  ->  A. x  -.  A. y  -.  ph )
61, 5hbxfrbi 1558 1  |-  ( E. y ph  ->  A. x E. y ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1530   E.wex 1531
This theorem is referenced by:  19.12  1746
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727
This theorem depends on definitions:  df-bi 177  df-ex 1532
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