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Theorem bj-rexcom4b 31481
Description: Remove from rexcom4b 3069 dependency on ax-ext 2431 and ax-13 2091 (and on df-or 372, df-cleq 2444, df-nfc 2581, df-v 3047). The hypothesis uses  V instead of  _V (see bj-isseti 31473 for the motivation). Use bj-rexcom4bv 31480 instead when sufficient (in particular when  V is substituted for  _V). (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
bj-rexcom4b.1  |-  B  e.  V
Assertion
Ref Expression
bj-rexcom4b  |-  ( E. x E. y  e.  A  ( ph  /\  x  =  B )  <->  E. y  e.  A  ph )
Distinct variable groups:    x, A    x, B    x, y    ph, x
Allowed substitution hints:    ph( y)    A( y)    B( y)    V( x, y)

Proof of Theorem bj-rexcom4b
StepHypRef Expression
1 bj-rexcom4a 31479 . 2  |-  ( E. x E. y  e.  A  ( ph  /\  x  =  B )  <->  E. y  e.  A  (
ph  /\  E. x  x  =  B )
)
2 bj-rexcom4b.1 . . . . 5  |-  B  e.  V
32bj-isseti 31473 . . . 4  |-  E. x  x  =  B
43biantru 508 . . 3  |-  ( ph  <->  (
ph  /\  E. x  x  =  B )
)
54rexbii 2889 . 2  |-  ( E. y  e.  A  ph  <->  E. y  e.  A  (
ph  /\  E. x  x  =  B )
)
61, 5bitr4i 256 1  |-  ( E. x E. y  e.  A  ( ph  /\  x  =  B )  <->  E. y  e.  A  ph )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188    /\ wa 371    = wceq 1444   E.wex 1663    e. wcel 1887   E.wrex 2738
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-11 1920  ax-12 1933
This theorem depends on definitions:  df-bi 189  df-an 373  df-ex 1664  df-sb 1798  df-clab 2438  df-clel 2447  df-rex 2743
This theorem is referenced by: (None)
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