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Theorem hbe1w 1867
Description: Weak version of hbe1 1891. See comments for ax10w 1877. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.)
Hypothesis
Ref Expression
hbn1w.1  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
hbe1w  |-  ( E. x ph  ->  A. x E. x ph )
Distinct variable groups:    ph, y    ps, x    x, y
Allowed substitution hints:    ph( x)    ps( y)

Proof of Theorem hbe1w
StepHypRef Expression
1 df-ex 1660 . 2  |-  ( E. x ph  <->  -.  A. x  -.  ph )
2 hbn1w.1 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
32notbid 295 . . 3  |-  ( x  =  y  ->  ( -.  ph  <->  -.  ps )
)
43hbn1w 1865 . 2  |-  ( -. 
A. x  -.  ph  ->  A. x  -.  A. x  -.  ph )
51, 4hbxfrbi 1690 1  |-  ( E. x ph  ->  A. x E. x ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187   A.wal 1435   E.wex 1659
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841
This theorem depends on definitions:  df-bi 188  df-ex 1660
This theorem is referenced by: (None)
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