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Theorem hbe1w 1723
Description: Weak version of hbe1 1746. See comments for ax6w 1732. 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 1551 . 2  |-  ( E. x ph  <->  -.  A. x  -.  ph )
2 hbn1w.1 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
32notbid 286 . . 3  |-  ( x  =  y  ->  ( -.  ph  <->  -.  ps )
)
43hbn1w 1721 . 2  |-  ( -. 
A. x  -.  ph  ->  A. x  -.  A. x  -.  ph )
51, 4hbxfrbi 1577 1  |-  ( E. x ph  ->  A. x E. x ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177   A.wal 1549   E.wex 1550
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551
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