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Theorem hbex 1853
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 1548 . 2  |-  ( E. y ph  <->  -.  A. y  -.  ph )
2 hbex.1 . . . . 5  |-  ( ph  ->  A. x ph )
32hbn 1784 . . . 4  |-  ( -. 
ph  ->  A. x  -.  ph )
43hbal 1743 . . 3  |-  ( A. y  -.  ph  ->  A. x A. y  -.  ph )
54hbn 1784 . 2  |-  ( -. 
A. y  -.  ph  ->  A. x  -.  A. y  -.  ph )
61, 5hbxfrbi 1574 1  |-  ( E. y ph  ->  A. x E. y ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1546   E.wex 1547
This theorem is referenced by:  nfex  1855  19.12OLD  1860
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753
This theorem depends on definitions:  df-bi 178  df-ex 1548
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