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Theorem bj-cbvexv 34644
Description: Version of cbvex 2029 with a dv condition, which does not require ax-13 2006. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-cbvalv.1  |-  F/ y
ph
bj-cbvalv.2  |-  F/ x ps
bj-cbvalv.3  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
bj-cbvexv  |-  ( E. x ph  <->  E. y ps )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    ps( x, y)

Proof of Theorem bj-cbvexv
StepHypRef Expression
1 bj-cbvalv.1 . . . . 5  |-  F/ y
ph
21nfn 1909 . . . 4  |-  F/ y  -.  ph
3 bj-cbvalv.2 . . . . 5  |-  F/ x ps
43nfn 1909 . . . 4  |-  F/ x  -.  ps
5 bj-cbvalv.3 . . . . 5  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
65notbid 292 . . . 4  |-  ( x  =  y  ->  ( -.  ph  <->  -.  ps )
)
72, 4, 6bj-cbvalv 34643 . . 3  |-  ( A. x  -.  ph  <->  A. y  -.  ps )
87notbii 294 . 2  |-  ( -. 
A. x  -.  ph  <->  -. 
A. y  -.  ps )
9 df-ex 1621 . 2  |-  ( E. x ph  <->  -.  A. x  -.  ph )
10 df-ex 1621 . 2  |-  ( E. y ps  <->  -.  A. y  -.  ps )
118, 9, 103bitr4i 277 1  |-  ( E. x ph  <->  E. y ps )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184   A.wal 1397   E.wex 1620   F/wnf 1624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862
This theorem depends on definitions:  df-bi 185  df-ex 1621  df-nf 1625
This theorem is referenced by:  bj-cbvexvv  34646  bj-axrep1  34721  bj-axrep2  34722  bj-axrep4  34724
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