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Theorem hbex 1568
Description: If  x is not free in  ph, it is not free in  E. y ph. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.)
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 hbe1 1425 . . 3  |-  ( E. y ph  ->  A. y E. y ph )
21hbal 1407 . 2  |-  ( A. x E. y ph  ->  A. y A. x E. y ph )
3 hbex.1 . . 3  |-  ( ph  ->  A. x ph )
4 19.8a 1523 . . 3  |-  ( ph  ->  E. y ph )
53, 4alrimih 1399 . 2  |-  ( ph  ->  A. x E. y ph )
62, 5exlimih 1525 1  |-  ( E. y ph  ->  A. x E. y ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1283   E.wex 1422
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-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  nfex  1569  excomim  1594  19.12  1596  cbvexh  1680  cbvexdh  1844  hbsbv  1860  hbeu1  1953  hbmo  1982  moexexdc  2027
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