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Theorem bj-rexcom4bv 34848
Description: Version of bj-rexcom4b 34849 with a dv condition on  x ,  V, hence removing dependency on df-sb 1745 and df-clab 2440 (so that it depends on df-clel 2449 and df-rex 2810 only on top of first-order logic). Prefer its use over bj-rexcom4b 34849 when sufficient (in particular when  V is substituted for  _V). (Contributed by BJ, 14-Sep-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
bj-rexcom4bv.1  |-  B  e.  V
Assertion
Ref Expression
bj-rexcom4bv  |-  ( E. x E. y  e.  A  ( ph  /\  x  =  B )  <->  E. y  e.  A  ph )
Distinct variable groups:    x, A    x, B    x, V    x, y    ph, x
Allowed substitution hints:    ph( y)    A( y)    B( y)    V( y)

Proof of Theorem bj-rexcom4bv
StepHypRef Expression
1 bj-rexcom4a 34847 . 2  |-  ( E. x E. y  e.  A  ( ph  /\  x  =  B )  <->  E. y  e.  A  (
ph  /\  E. x  x  =  B )
)
2 bj-rexcom4bv.1 . . . . 5  |-  B  e.  V
32bj-issetiv 34840 . . . 4  |-  E. x  x  =  B
43biantru 503 . . 3  |-  ( ph  <->  (
ph  /\  E. x  x  =  B )
)
54rexbii 2956 . 2  |-  ( E. y  e.  A  ph  <->  E. y  e.  A  (
ph  /\  E. x  x  =  B )
)
61, 5bitr4i 252 1  |-  ( E. x E. y  e.  A  ( ph  /\  x  =  B )  <->  E. y  e.  A  ph )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367    = wceq 1398   E.wex 1617    e. wcel 1823   E.wrex 2805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-11 1847
This theorem depends on definitions:  df-bi 185  df-an 369  df-ex 1618  df-clel 2449  df-rex 2810
This theorem is referenced by: (None)
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